openlm-research/open_llama_3b
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\section{Introduction}
Let $G$ be a simple undirected graph with the \textit{vertex set} $V(G)$ and the \textit{edge set} $E(G)$. A vertex with degree one is called a \textit{pendant vertex}. The distance between the vertices $u$ and $v$ in graph $G$ is denoted by $d_G(u,v)$. A cycle $C$ is called \textit{chordless} if $C$ has no \textit{cycle chord} (that is an edge not in the edge set of $C$ whose endpoints lie on the vertices of $C$).
The \textit{Induced subgraph} on vertex set $S$ is denoted by $\langle S\rangle$. A path that starts in $v$ and ends in $u$ is denoted by $\stackrel\frown{v u}$.
A \textit{traceable} graph is a graph that possesses a Hamiltonian path.
In a graph $G$, we say that a cycle $C$ is \textit{formed by the path} $Q$ if $ | E(C) \setminus E(Q) | = 1 $. So every vertex of $C$ belongs to $V(Q)$.
In 2011 the following conjecture was proposed:
\begin{conjecture}(Hoffmann-Ostenhof \cite{hoffman})
Let $G$ be a connected cubic graph. Then $G$ has a decomposition into a spanning tree, a matching and a family of cycles.
\end{conjecture}
Conjecture \theconjecture$\,$ also appears in Problem 516 \cite{cameron}. There are a few partial results known for Conjecture \theconjecture. Kostochka \cite{kostocha} noticed that the Petersen graph, the prisms over cycles, and many other graphs have a decomposition desired in Conjecture \theconjecture. Ozeki and Ye \cite{ozeki} proved that the conjecture holds for 3-connected cubic plane graphs. Furthermore, it was proved by Bachstein \cite{bachstein} that Conjecture \theconjecture$\,$ is true for every 3-connected cubic graph embedded in torus or Klein-bottle. Akbari, Jensen and Siggers \cite[Theorem 9]{akbari} showed that Conjecture \theconjecture$\,$ is true for Hamiltonian cubic graphs.
In this paper, we show that Conjecture \theconjecture$\,$ holds for traceable cubic graphs.
\section{Results}
Before proving the main result, we need the following lemma.
\begin{lemma}
\label{lemma:1}
Let $G$ be a cubic graph. Suppose that $V(G)$ can be partitioned into a tree $T$ and finitely many cycles such that there is no edge between any pair of cycles (not necessarily distinct cycles), and every pendant vertex of $T$ is adjacent to at least one vertex of a cycle. Then, Conjecture \theconjecture$\,$ holds for $G$.
\end{lemma}
\begin{proof}
By assumption, every vertex of each cycle in the partition is adjacent to exactly one vertex of $T$. Call the set of all edges with one endpoint in a cycle and another endpoint in $T$ by $Q$.
Clearly, the induced subgraph on $E(T) \cup Q$ is a spanning tree of $G$. We call it $T'$. Note that every edge between a pendant vertex of $T$ and the union of cycles in the partition is also contained in $T'$. Thus, every pendant vertex of $T'$ is contained in a cycle of the partition. Now, consider the graph $H = G \setminus E(T')$. For every $v \in V(T)$, $d_H(v) \leq 1$. So Conjecture \theconjecture$\,$ holds for $G$. \vspace{1em}
\end{proof}
\noindent\textbf{Remark 1.}
\label{remark:1}
Let $C$ be a cycle formed by the path $Q$. Then clearly there exists a chordless cycle formed by $Q$.
Now, we are in a position to prove the main result.
\begin{theorem}
Conjecture \theconjecture$\,$ holds for traceable cubic graphs.
\end{theorem}
\begin{proof}
Let $G$ be a traceable cubic graph and $P : v_1, \dots, v_n$ be a Hamiltonian path in $G$. By \cite[Theorem 9]{akbari}, Conjecture A holds for $v_1 v_n \in E(G)$. Thus we can assume that $v_1 v_n \notin E(G)$. Let $v_1 v_j, v_1 v_{j'}, v_i v_n, v_{i'} v_n \in E(G)\setminus E(P)$ and $j' < j < n$, $1 < i < i'$. Two cases can occur:
\begin{enumerate}[leftmargin=0pt,label=]
\item
\textbf{Case 1.}
Assume that $i < j$. Consider the following graph in Figure \ref{fig:overlapping} in which the thick edges denote the path $P$. Call the three paths between $v_j$ and $v_i$, from the left to the right, by $P_1$, $P_2$ and $P_3$, respectively (note that $P_1$ contains the edge $e'$ and $P_3$ contains the edge $e$).
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/overlapping.pdf}
\caption{Paths $P_1$, $P_2$ and $P_3$}
\label{fig:overlapping}
\end{center}
\end{figure}
If $P_2$ has order $2$, then $G$ is Hamiltonian and so by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds. Thus we can assume that $P_1$, $P_2$ and $P_3$ have order at least $3$. Now, consider the following subcases:\\
\begin{enumerate}[leftmargin=0pt,label=]
\label{case:1}
\item \textbf{Subcase 1.} There is no edge between $V(P_r)$ and $V(P_s)$ for $1 \leq r < s \leq 3$. Since every vertex of $P_i$ has degree 3 for every $i$, by \hyperref[remark:1]{Remark 1}$\,$ there are two chordless cycles $C_1$ and $C_2$ formed by $P_1$ and $P_2$, respectively.
Define a tree $T$ with the edge set
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\item \textbf{Subcase 2.}
\label{case:edge}
There exists at least one edge between some $P_r$ and $P_s$, $r<s$. With no loss of generality, assume that $r=1$ and $s=2$. Suppose that $ab \in E(G)$, where $a \in V(P_1)$, $b \in V(P_2)$ and $d_{P_1}(v_j, a) + d_{P_2}(v_j, b)$ is minimum.
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/ab.pdf}
\caption{The edge $ab$ between $P_1$ and $P_2$}
\label{fig:ab}
\end{center}
\end{figure}
Three cases occur: \\
(a) There is no chordless cycle formed by either of the paths $\stackrel\frown{v_j a}$ or $\stackrel\frown{v_j b}$. Let $C$ be the chordless cycle $\stackrel\frown{v_j a}\stackrel\frown{ b v_j}$. Define $T$ with the edge set
$$ E\Big(\langle V(G) \setminus V(C)\rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C\}$. \\
(b) There are two chordless cycles, say $C_1$ and $C_2$, respectively formed by the paths $\stackrel\frown{v_j a}$ and $\stackrel\frown{v_j b}$. Now, consider the partition $C_1$, $C_2$ and the tree induced on the following edges,
$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \; \bigcap \; E\Big(\bigcup_{i=1}^3 P_i\Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
(c) With no loss of generality, there exists a chordless cycle formed by the path $\stackrel\frown{v_j a}$ and there is no chordless cycle formed by the path $\stackrel\frown{v_j b}$.
First, suppose that for every chordless cycle $C_t$ on $\stackrel\frown{v_j a}$, at least one of the vertices of $C_t$ is adjacent to a vertex in $V(G) \setminus V(P_1)$.
We call one of the edges with one end in $C_t$ and other endpoint in $V(G) \setminus V(P_1)$ by $e_t$. Let $v_j=w_0, w_1, \dots, w_l=a$ be all vertices of the path $\stackrel\frown{v_j a}$ in $P_1$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$.
Define a tree $T$ whose edge set is the thin edges in Figure \ref{fig:deltaCycle}.\\
Call the cycle $w_0 w_{i_1} \dots w_l \stackrel\frown{b w_0}$ by $C'$. Now, by removing $C'$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are contained in $\stackrel\frown{v_j a}$ remain. Note that there exists a path of order $2$ in $C'$ which by adding this path to $Q_i$ we find a cycle $C_{t_i}$, for some $i$. Hence there exists an edge $e_{t_i}$ connecting $Q_i$ to $V(G) \setminus V(P_1)$. Now, we define a tree $T$ whose the edge set is,
$$\quad\quad\quad \bigg( E\Big(\langle V(G) \setminus V(C') \rangle \Big)\; \bigcap \; \Big(\bigcup_{i=1}^3 E(P_i)\Big) \bigg) \bigcup \Big(\big\{e_{t_i} \mid 1 \leq i \leq q \big\} \Big).$$
Apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C'\}$.\\
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/deltaCycle.pdf}
\caption{The cycle $C'$ and the tree $T$}
\label{fig:deltaCycle}
\end{center}
\end{figure}
Next, assume that there exists a cycle $C_1$ formed by $\stackrel\frown{v_j a}$ such that none of the vertices of $C_1$ is adjacent to $V(G) \setminus V(P_1)$. Choose the smallest cycle with this property. Obviously, this cycle is chordless. Now, three cases can be considered:\\
\begin{enumerate}[leftmargin=5pt,label=(\roman*)]
\item There exists a cycle $C_2$ formed by $P_2$ or $P_3$. Define the partition $C_1$, $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is at least one edge between $V(P_2)$ and $V(P_3)$. Let $ab \in E(G)$, $a \in V(P_2)$ and $b \in V(P_3)$ and moreover $d_{P_2}(v_j, a) + d_{P_3}(v_j,b)$ is minimum. Notice that the cycle $\stackrel\frown{v_j a} \stackrel\frown{b v_j}$ is chordless. Let us call this cycle by $C_2$. Now, define the partition $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is no edge between $V(P_2)$ and $V(P_3)$. Let $C_2$ be the cycle consisting of two paths $P_2$ and $P_3$. Define the partition $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.
\end{enumerate}
\end{enumerate}
\vspace{5mm}
\item
\textbf{Case 2.}
\label{case:2}
Assume that $j < i$ for all Hamiltonian paths. Among all Hamiltonian paths consider the path such that $i'-j'$ is maximum. Now, three cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=]
\item \textbf{Subcase 1.} There is no $s < j'$ and $t > i'$ such that $v_s v_t \in E(G)$. By \hyperref[remark:1]{Remark 1} $\,$ there are two chordless cycles $C_1$ and $C_2$, respectively formed by the paths $v_1 v_{j'}$ and $v_{i'} v_n$. By assumption there is no edge $xy$, where $x \in V(C_1)$ and $y \in V(C_2)$.
Define a tree $T$ with the edge set:
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_{i'}v_n, v_{j'}v_1\} \Big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\item \textbf{Subcase 2.}
\label{subcase:22} There are at least four indices $s, s' < j$ and $t, t' > i$ such that $v_s v_t, v_{s'} v_{t'} \in E(G)$. Choose four indices $g, h < j$ and $e, f > i$ such that $v_h v_e, v_g v_f \in E(G)$ and $|g-h| + |e-f|$ is minimum.
\begin{figure}[H]
\begin{center}
\includegraphics[width=90mm]{engImages/case2-subcase2.pdf}
\caption{Two edges $v_h v_e$ and $v_g v_f$}
\label{fig:non-overlapping}
\end{center}
\end{figure}
Three cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=(\alph*)]
\item There is no chordless cycle formed by $\stackrel\frown{v_g v_h}$ and by $\stackrel\frown{v_e v_f}$.
Consider the cycle $\stackrel\frown{v_g v_h} \stackrel\frown{v_e v_f}v_g$ and call it $C$. Now, define a tree $T$ with the edge set,
$$\,\,\,E\Big(\langle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$
apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C\}$.\\
\item With no loss of generality, there exists a chordless cycle formed by $\stackrel\frown{v_e v_f}$ and there is no chordless cycle formed by the path $\stackrel\frown{v_g v_h}$. First suppose that there is a chordless cycle $C_1$ formed by $\stackrel\frown{v_e v_f}$ such that there is no edge between $V(C_1)$ and $\{v_1, \dots, v_j\}$. By \hyperref[remark:1]{Remark 1} $,$ there exists a chordless cycle $C_2$ formed by $\stackrel\frown{v_1 v_j}$. By assumption there is no edge between $V(C_1)$ and $V(C_2)$. Now, define a tree $T$ with the edge set,
$$\quad\quad\quad\quad E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.
$\;$ Next assume that for every cycle $C_r$ formed by $\stackrel\frown{v_e v_f}$, there are two vertices $x_r \in V(C_r)$ and $y_r \in \{v_1, \dots, v_j\}$ such that $x_r y_r \in E(G)$. Let $v_e=w_0, w_1, \dots, w_l=v_f$ be all vertices of the path $\stackrel\frown{v_e v_f}$ in $P$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$. Consider the cycle $w_0 w_{i_1} \dots w_l \stackrel\frown{v_g v_h}$ and call it $C$. Now, by removing $C$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are contained in $\stackrel\frown{v_e v_f}$ remain. Note that there exists a path of order $2$ in $C$ which by adding this path to $Q_i$ we find a cycle $C_{r_i}$, for some $i$. Hence there exists an edge $x_{r_i} y_{r_i}$ connecting $Q_i$ to $V(G) \setminus V(\stackrel\frown{v_e v_f})$. We define a tree $T$ whose edge set is the edges,
$$\quad\quad\quad\quad\quad\quad E\Big(\langle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \cup \big\{x_{r_i} y_{r_i} \mid 1 \leq i \leq q\big\} \Big),$$
then apply \hyperref[lemma:1]{Lemma 1} $\,$ on the partition $\{T, C\}$.\\
\begin{figure}[H]
\begin{center}
\includegraphics[width=90mm]{engImages/deltaNonOverlapping.pdf}
\caption{The tree $T$ and the shortest path $w_0 w_{i_1}\dots w_l$}
\label{fig:delta-non-overlapping}
\end{center}
\end{figure}
\item There are at least two chordless cycles, say $C_1$ and $C_2$ formed by the paths $\stackrel\frown{v_g v_h}$ and $\stackrel\frown{v_e v_f}$, respectively. Since $|g-h| + |e-f|$ is minimum, there is no edge $xy \in E(G)$ with $x \in V(C_1)$ and $y \in V(C_2)$. Now, define a tree $T$ with the edge set,
$$\quad\quad\quad\quad E\Big( \langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_1 v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\end{enumerate}
\item \textbf{Subcase 3.} There exist exactly two indices $s,t$, $s < j' < i' < t$ such that $v_s v_t \in E(G)$ and there are no two other indices $s', t'$ such that $s' < j < i < t'$ and $v_{s'} v_{t'} \in E(G)$. We can assume that there is no cycle formed by $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$, to see this by symmetry consider a cycle $C$ formed by $\stackrel\frown{v_{s+1} v_j}$. By \hyperref[remark:1]{Remark 1} $\,$ there exist chordless cycles $C_1$ formed by $\stackrel\frown{v_{s+1} v_j}$ and $C_2$ formed by $\stackrel\frown{v_{i} v_n}$. By assumption $v_s v_t$ is the only edge such that $s < j$ and $t > i \;$. Therefore, there is no edge between $V(C_1)$ and $V(C_2)$. Now, let $T$ be a tree defined by the edge set,
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.\\
$\quad$Furthermore, we can also assume that either $s \neq j'-1$ or $t \neq i'+1$, otherwise we have the Hamiltonian cycle $\stackrel\frown{v_1 v_s} \stackrel\frown{v_t v_n} \stackrel\frown{v_{i'} v_{j'}} v_1$ and by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds.
$\quad$By symmetry, suppose that $s \neq j'-1$. Let $v_k$ be the vertex adjacent to $v_{j'-1}$, and $k \notin \{j'-2, j'\}$. It can be shown that $k > j'-1$, since otherwise by considering the Hamiltonian path $P': \; \stackrel\frown{ v_{k+1} v_{j'-1}}\stackrel\frown{v_k v_1} \stackrel\frown{v_{j'} v_n}$, the new $i'-j'$ is greater than the old one and this contradicts our assumption about $P$ in the \hyperref[case:2]{Case 2}.
$\quad$We know that $j' < k < i$. Moreover, the fact that $\stackrel\frown{v_{s+1} v_j}$ does not form a cycle contradicts the case that $j' < k \le j$. So $j < k < i$. Consider two cycles $C_1$ and $C_2$, respectively with the vertices $v_1 \stackrel\frown{v_{j'} v_{j}} v_1$ and $v_n \stackrel\frown{v_{i'} v_{i}} v_n$. The cycles $C_1$ and $C_2$ are chordless, otherwise there exist cycles formed by the paths $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$. Now, define a tree $T$ with the edge set
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_s v_t, v_k v_{j'-1}\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.
\end{enumerate}
\end{enumerate}
\end{proof}
\noindent\textbf{Remark 2.}
\label{remark:2}
Indeed, in the proof of the previous theorem we showed a stronger result, that is, for every traceable cubic graph there is a decomposition with at most two cycles.
| {'timestamp': '2016-07-19T02:04:55', 'yymm': '1607', 'arxiv_id': '1607.04768', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04768'} | arxiv |
\section{Principle of nano strain-amplifier}
\begin{figure*}[t!]
\centering
\includegraphics[width=5.4in]{Fig1}
\vspace{-0.5em}
\caption{Schematic sketches of nanowire strain sensors. (a)(b) Conventional non-released and released NW structure;
(c)(d) The proposed nano strain-amplifier and its simplified physical model.}
\label{fig:fig1}
\vspace{-1em}
\end{figure*}
Figure \ref{fig:fig1}(a) and 1(b) show the concept of the conventional structures of piezoresistive sensors. The piezoresistive elements are either released from, or kept on, the substrate. The sensitivity ($S$) of the sensors is defined based on the ratio of the relative resistance change ($\Delta R/R$) of the sensing element and the strain applied to the substrate ($\varepsilon_{sub}$):
\begin{equation}
S = (\Delta R/R)/\varepsilon_{sub}
\label{eq:sensitivity}
\end{equation}
In addition, the relative resistance change $\Delta R/R$ can be calculated from the gauge factor ($GF$) of the material used to make the piezoresistive elements: $\Delta R/R = GF \varepsilon_{ind}$, where $\varepsilon_{ind}$ is the strain induced into the piezoresistor. In most of the conventional strain gauges as shown in Fig. \ref{fig:fig1} (a,b), the thickness of the sensing layer is typically below a few hundred nanometers, which is much smaller than that of the substrate. Therefore, the strain induced into the piezoresistive elements is approximately the same as that of the substrate ($\varepsilon_{ind} \approx \varepsilon_{sub}$). Consequently, to improve the sensitivity of strain sensors (e.g. enlarging $\Delta R/R$), electrical approaches which can enlarge the gauge factor ($GF$) are required. Nevertheless, as aforementioned, the existence of the large gauge factor in nanowires due to quantum confinement or surface state, is still considered as controversial.
It is also evident from Eq. \ref{eq:sensitivity} that the sensitivity of strain sensors can also be improved using a mechanical approach, which enlarges the strain induced into the piezoresistive element. Figure \ref{fig:fig1}(c) shows our proposed nano strain-amplifier structure, in which the piezoresistive nanowires are locally fabricated at the centre of a released bridge. The key idea of this structure is that, under a certain strain applied to the substrate, a large strain will be concentrated at the locally fabricated SiC nanowires. The working principle of the nano strain-amplifier is similar to that of the well-known dogbone structure, which is widely used to characterize the tensile strength of materials \cite{dogbone1,dogbone2}. That is, when a stress is applied to the dogbone-shape of a certain material, a crack, if generated, will occur at the middle part of the dogbone. The large strain concentrated at the narrow area located at the centre part with respect to the wider areas located at outer region, causes the crack. Qualitative and quantitative explanations of the nano strain-amplifier are presented as follows.
For the sake of simplicity, the released micro frame and nanowire (single wire or array) of the nano strain-amplifier can be considered as solid springs, Fig. \ref{fig:fig1}(d). The stiffness of these springs are proportional to their width ($w$) and inversely proportional to their length (l): $K \propto w/l$. Consequently, the model of the released nanowire and micro frames can be simplified as a series of springs, where the springs with higher stiffness correspond to the micro frame, and the single spring with lower stiffness corresponds to the nanowire. It is well-known in classical physics that, for serially connected springs, a larger strain will be concentrated in the low--stiffness string, while a smaller strain will be induced in the high--stiffness string \cite{Springbook}. The following analysis quantitatively explained the amplification of the strain.
\begin{figure}[b!]
\centering
\includegraphics[width=3in]{Fig2}
\vspace{-1em}
\caption{Finite element analysis of the strain induced in to the nanowire array utilizing nano strain-amplifier.}
\label{fig:fig2}
\end{figure}
When a tensile mechanical strain ($\varepsilon_{sub}$) is applied to the substrate, the released structure will also be elongated. Since the stiffness of the released frame is much smaller than that of the substrate, it is safe to assume that the released structure will follows the elongation of the substrate. The displacement of the released structure $\Delta L$ is:
\begin{equation}
\Delta L = \Delta L_m + \Delta L_n = L_m \varepsilon_m + L_n \varepsilon_n
\label{eq:displacement}
\end{equation}
where $L_m$, $L_n$ are the length; $\Delta L_m$, $\Delta L_n$ are the displacement; and $\varepsilon_m$, $\varepsilon_n$ are the strains induced into the micro spring and nano spring, respectively. The subscripts m and n stand for the micro frames and nanowires, respectively. Furthermore, due to the equilibrium of the stressing force ($F$) along the series of springs, the following relationship is established: $F= K_m\Delta L_m = K_n \Delta L_n$, where $K_m$, $K_n$ are the stiffness of the released micro frames and nanowires, respectively. Consequently the relationship between the displacement of the micro frame (higher stiffness) and nanowires (lower stiffness) is:
\begin{equation}
\frac{\Delta L_m}{\Delta L_n}=\frac{K_n}{K_m}=\frac{L_mw_n}{L_nw_m}
\label{eq:euili}
\end{equation}
Substituting Eqn. \ref{eq:euili} into Eqn. \ref{eq:displacement}, the strain induced into the locally fabricated nanowires is:
\begin{equation}
\varepsilon_n = \frac{\Delta L_n}{L_n} = \frac{1}{1-\frac{w_m-w_n}{w_m}\frac{L_m}{L}}\varepsilon_{sub}
\label{eq:strainamp}
\end{equation}
Equation \ref{eq:strainamp} indicates that increasing the ratio of $w_m/w_n$ and $L_m/L_n$ significantly amplifies the strain induced into the nanowire from the strain applied to the substrate. This model is also applicable to the case of nanowire arrays, in which $w_n$ is the total width of all nanowires in the array.
The theoretical model is then verified using the finite element analysis (FEA). In the FEA simulation, we compare the strain induced into (i) non released nanowires, (ii) the conventionally released nanowires, and (iii) our nano strain-amplifier structure, using COMSOL Multiphysics \texttrademark. In our nano strain amplifying structure, the width of the released frame was set to be 8 $\mu$m, while the width of each nanowire in the array (3 wires) was set to be 370 nm. The nanowires array structure was selected as it can enhance the electrical conductance of the SiC nanowires resistor which makes the subsequent experimental demonstration easier. The ratio between the length of nanowires and micro bridge was set to be 1: 20. With this geometrical dimensions, strain induced into nanowires array $\varepsilon_n$ was numerically calculated to be approximately 6 times larger than $\varepsilon_{sub}$, Eqn. \ref{eq:strainamp}. The simulation results show that for all structure, the elongation of non-released and released nanowires follow that of the substrate. In addition, strain was almost completely transferred into conventional released and non-released structures. Furthermore, the ratio of the strain induced in to the locally fabricated nanowires was estimated to be 5.9 times larger than that of the substrate, Fig. \ref{fig:fig2}. These results are in solid agreement with the theoretical analysis presented above. For a nanowire array with an average width of 470 nm, the amplified gain of strain was found to be 4.5.
Based on the theoretical analysis, we conducted the following experiments to demonstrate the high sensitivity of SiC nanowire strain sensors using the nano strain-amplifier. A thin 3C-SiC film with its thickness of 300 nm was epitaxially grown on a 150 mm diameter Si wafer using low pressure chemical vapour deposition \cite{SiC_growth}. The film was \emph{in situ} doped using Al dopants. The carrier concentration of the p-type 3C-SiC was found to be $5 \times 10^{18}$ cm$^{-3}$, using a hot probe technique \cite{philip}. The details of the characteristics of the grown film can be found elsewhere \cite{Phan_JMC}. Subsequently, I-shape p-type SiC resistors with aluminum electrodes deposited on the surface were patterned using inductive coupled plasma (ICP) etching. As the piezoresistance of p-type 3C-SiC depends on crystallographic orientation, all SiC resistors of the present work were aligned along [110] direction to maximize the piezoresistive effect. Next, the micro scale SiC resistors were then released from the Si substrate using dry etching (XeF$_2$). Finally, SiC nanowire arrays were formed at the centre of the released bridge using focused ion beam (FIB). Two types of nanowire array were fabricated with three nanowires for each array. The average width of each nanowire in each type were 380 nm and 470 nm, respectively. Figure \ref{fig:fig3} shows the SEM images of the fabricated samples, including the conventional released structure, non-released nanowires, and the nano strain-amplifier.
\begin{figure}[t!]
\centering
\includegraphics[width=3in]{Fig3}
\caption{SEM image of SiC strain sensors. (a) Released SiC micro bridge used for the subsequent fabrication of the nano strain-amplifier; (b) SEM of a micro SiC resistor where the SiC nanowires array were formed using FIB; (c) SEM of non-released SiC nanowires; (d) SEM of locally fabricated SiC nanowires released from the Si substrate (nano strain-amplifier).}
\label{fig:fig3}
\vspace{-1em}
\end{figure}
The current voltage (I-V) curves of all fabricated samples were characterized using a HP 4145 \texttrademark ~parameter analyzer. The linear relationship between the applied voltage and measured current, indicated that Al made a good Ohmic contact with the highly doped SiC resistance, Fig. \ref{fig:IV}. Additionally, the electrical conductivity of both nanowires and micro frame estimated from the I-V curve and the dimensions of the resistors shows almost the same value. This indicated that the FIB process did not cause a significant surface damage to the fabricated nanowires.
\begin{figure}[b!]
\centering
\includegraphics[width=3in]{Fig4}
\vspace{-1.5em}
\caption{Current voltage curves of the fabricated SiC resistors.}
\label{fig:IV}
\end{figure}
The bending experiment was used to characterize the piezoresistive effect in micro size SiC resistors and locally fabricated SiC nanowire array. In this experiment one end of the Si cantilever (with a thickness of 625 $\mu$m, and a width of 7 mm) was fixed while the other end was deflected by applying different forces. The distance from the fabricated nanowires to the free end of the Si cantilever was approximately 45 mm. The strain induced into the Si substrate is $\varepsilon_\text{sub} = Mt/2EI$, where $M$ is the applied bending moment; and $t$, $E$ and $I$ are the thickness, Young's modulus and the moment of inertia of the Si cantilever, respectively. The response of the SiC resistance to applied strain was then measured using a multimeter (Agilent \texttrademark 34401 A).
\begin{figure}[h!]
\centering
\includegraphics[width=3in]{Fig5.eps}
\vspace{-1.5em}
\caption{Experimental results. (a) A comparision between the relative resistance change in the nano strain-amplifiers, non released nanowires and released micro frames; (b) The repeatability of the SiC nanowires strain sensors utilizing the proposed structure.}
\label{fig:DRR}
\vspace{-1em}
\end{figure}
The relative resistance change ($\Delta R/R$) of the micro and nano SiC resistors was plotted against the strain induced into the Si substrate $\varepsilon_{sub}$, Fig. \ref{fig:DRR}(a). For all fabricated samples, the relative resistance change shows a good linear relationship with the applied strain ($\varepsilon_{sub}$). In addition, with the same applied strain to the Si substrate, the resistance change of the SiC nanowires using the nano strain-amplifier was much larger than that of the the SiC micro resistor and the conventional non-released SiC nanowires. In addition, reducing the width of the SiC nanowires also resulted in the increase of the sensitivity. The magnitude of the piezoresistive effect in the nano strain-amplifier as well as conventional structures were then quantitatively evaluated based on the effective gauge factor ($GF_{eff}$), which is defined as the ratio of the relative resistance change to the applied strain to the substrate: $GF_{eff} = (\Delta R/R)/\varepsilon_{sub}$. Accordingly, the effective gauge factor of the released micro SiC was found to be 28, while that of the non-released SiC nanowires was 35. From the data shown in Fig. \ref{fig:DRR}, the effective gauge factor of the 380 nm and 470 nm SiC nanowires in the nano strain-amplifier were calculated as 150 and 124, respectively. Thus for nanowire arrays with average widths of 380 nm and 470 nm, the sensitivity of the nano strain-amplifier was 5.4 times and 4.6 times larger than the bulk SiC, respectively. These results were consistent with analytical and numerical models presented above. The relative resistance change of the nano strain-amplifier also showed excellent linearity with the applied strain, with a linear regression of above 99\%.
The resistance change of the nano strain-amplifier can also be converted into voltage signals using a Wheatstone bridge, Fig. \ref{fig:DRR}(b). The output voltage of the nano strain-amplifier increases with increasing tensile strains from 0 ppm to 180 ppm, and returned to the initial value when the strain was completely removed, confirming a good repeatability after several strain induced cycles. The linearity of the relative resistance change, and the repeatability indicate that the proposed structure is promising for strain sensing applications.
In conclusion, this work presents a novel mechanical approach to obtain highly sensitive piezoresistance in nanowires based on a nano strain-amplifier. The key factor of the nano strain-amplifier lies on nanowires locally fabricated on a released micro structure. Experimental studies were conducted on SiC nanowires, confirming that by utilizing our nano strain-amplifier, the sensitivity of SiC nanowires was 5.4 times larger than that of conventional structures. This result indicated that the nano strain-amplifier is an excellent platform for ultra sensitive strain sensing applications.
| {'timestamp': '2016-07-18T02:07:38', 'yymm': '1607', 'arxiv_id': '1607.04531', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04531'} | arxiv |
\section{Introduction}\label{intro}
Gas has a fundamental role in shaping the evolution of galaxies,
through its accretion on to massive haloes, cooling and subsequent
fuelling of star formation, to the triggering of extreme luminous
activity around super massive black holes. Determining how the
physical state of gas in galaxies changes as a function of redshift is
therefore crucial to understanding how these processes evolve over
cosmological time. The standard model of the gaseous interstellar
medium (ISM) in galaxies comprises a thermally bistable medium
(\citealt*{Field:1969}) of dense ($n \sim 100$\,cm$^{-3}$) cold
neutral medium (CNM) structures, with kinetic temperatures of $T_{\rm
k} \sim 100$\,K, embedded within a lower-density ($n \sim
1$\,cm$^{-3}$) warm neutral medium (WNM) with $T_{\rm k} \sim
10^{4}$\,K. The WNM shields the cold gas and is in turn ionized by
background cosmic rays and soft X-rays (e.g. \citealt{Wolfire:1995,
Wolfire:2003}). A further hot ($T_{\rm k} \sim 10^{6}$\,K) ionized
component was introduced into the model by \cite{McKee:1977}, to
account for heating by supernova-driven shocks within the inter-cloud
medium. In the local Universe, this paradigm has successfully
withstood decades of observational scrutiny, although there is some
evidence (e.g. \citealt{Heiles:2003b}; \citealt*{Roy:2013b};
\citealt{Murray:2015}) that a significant fraction of the WNM may
exist at temperatures lower than expected for global conditions of
stability, requiring additional dynamical processes to maintain local
thermodynamic equilibrium.
Since atomic hydrogen (\mbox{H\,{\sc i}}) is one of the most abundant
components of the neutral ISM and readily detectable through either
the 21\,cm or Lyman $\alpha$ lines, it is often used as a tracer of
the large-scale distribution and physical state of neutral gas in
galaxies. The 21\,cm line has successfully been employed in surveying
the neutral ISM in the Milky Way
(e.g. \citealt{McClure-Griffiths:2009,Murray:2015}), the Local Group
(e.g. \citealt{Kim:2003,Bruns:2005,Braun:2009,Gratier:2010}) and
low-redshift Universe (see \citealt{Giovanelli:2016} for a
review). However, beyond $z \sim 0.4$ (\citealt{Fernandez:2016})
\mbox{H\,{\sc i}} emission from individual galaxies becomes too faint
to be detectable by current 21\,cm surveys and so we must rely on
absorption against suitably bright background radio (21\,cm) or UV
(Lyman-$\alpha$) continuum sources to probe the cosmological evolution
of \mbox{H\,{\sc i}}. The bulk of neutral gas is contained in
high-column-density damped Lyman-$\alpha$ absorbers (DLAs, $N_{\rm HI}
\geq 2 \times 10^{20}$\,cm$^{-2}$; see \citealt*{Wolfe:2005} for a
review), which at $z \gtrsim 1.7$ are detectable in the optical
spectra of quasars. Studies of DLAs provide evidence that the atomic
gas in the distant Universe appears to be consistent with a
multi-phase neutral ISM similar to that seen in the Local Group
(e.g. \citealt*{Lane:2000}; \citealt*{Kanekar:2001c};
\citealt*{Wolfe:2003b}). However, there is some variation in the cold
and warm fractions measured throughout the DLA population
(e.g. \citealt*{Howk:2005}; \citealt{Srianand:2005, Lehner:2008};
\citealt*{Jorgenson:2010}; \citealt{Carswell:2011, Carswell:2012,
Kanekar:2014a}; \citealt*{Cooke:2015}; \citealt*{Neeleman:2015}).
The 21-cm spin temperature affords us an important line-of-enquiry in
unraveling the physical state of high-redshift atomic gas. This
quantity is sensitive to the processes that excite the ground-state of
\mbox{H\,{\sc i}} in the ISM
(\citealt{Purcell:1956,Field:1958,Field:1959b,Bahcall:1969}) and
therefore dictates the detectability of the 21\,cm line in absorption.
In the CNM the spin temperature is governed by collisional excitation
and so is driven to the kinetic temperature, while the lower densities
in the WNM mean that the 21\,cm transition is not thermalized by
collisions between the hydrogen atoms, and so photo-excitation by the
background Ly $\alpha$ radiation field becomes important. Consequently
the spin temperature in the WNM is lower than the kinetic temperature,
in the range $\sim$1000 -- 5000\,K depending on the column density and
number of multi-phase components (\citealt{Liszt:2001}). Importantly,
the spin temperature measured from a single detection of extragalactic
absorption is equal to the harmonic mean of the spin temperature in
individual gas components, weighted by their column densities, thereby
providing a method of inferring the CNM fraction in high-redshift
systems.
Surveys for 21\,cm absorption in known redshifted DLAs have been used
to simultaneously measure the column density and spin temperature of
\mbox{H\,{\sc i}} (see \citealt{Kanekar:2014a} and references
therein). There is some evidence for an increase (at $4\,\sigma$
significance) in the spin temperature of DLAs at redshifts above $z =
2.4$, and a difference (at $6\,\sigma$ significance) between the
distribution of spin temperatures in DLAs and the Milky Way
(\citealt{Kanekar:2014a}). The implication that at least 90\,per\,cent
of high-redshift DLAs may have CNM fractions significantly less than
that measured for the Milky Way has important consequences for the
heating and cooling of neutral gas in the early Universe and star
formation (e.g. \citealt*{Wolfe:2003a}). However, these targeted
observations rely on the limited availability of simultaneous 21\,cm
and optical/UV data for the DLAs and assumes commonality between the
column density probed by the optical and radio sight-lines. The first
issue can be overcome by improving the sample statistics through
larger 21\,cm line surveys of high-redshift DLAs, but the latter
requires improvements to our methodology and understanding of the gas
distribution in these systems. There are also concerns about the
accuracy to which the fraction of the source structure subtended by
the absorber can be measured in each system, which can only be
resolved through spectroscopic very long baseline interferometry
(VLBI). It has been suggested that the observed evolution in spin
temperature could be biased by assumptions about the radio-source
covering factor (\citealt{Curran:2005}) and its behaviour as a
function of redshift (\citealt{Curran:2006b, Curran:2012b}).
In this paper we consider an approach using the statistical constraint
on the average spin temperature achievable with future large 21\,cm
surveys using precursor telescopes to the Square Kilometre Array
(SKA). This will enable independent verification of the evolution in
spin temperature at high redshift and provide a method of studying the
global properties of neutral gas below $z \approx 1.7$, where the
Lyman\,$\alpha$ line is inaccessible using ground-based observatories.
In an early attempt at a genuinely blind 21\,cm absorption survey,
\cite{Darling:2011} used pilot data from the Arecibo Legacy Fast
Arecibo L-band Feed Array (ALFALFA) survey to obtain upper limits on
the column density frequency distribution from 21\,cm absorption at
low redshift ($z \lesssim 0.06$). However, they also noted that the
number of detections could be used to make inferences about the ratio
of the spin temperature to covering factor. Building upon this work,
\cite{Wu:2015} found that their upper limits on the frequency
distribution function measured from the 40\,per\,cent ALFALFA survey
({$\alpha$}.40; \citealt{Haynes:2011}) could only be reconciled with
measurements from other low-redshift 21\,cm surveys if the typical
spin temperature to covering factor ratio was greater than 500\,K. At
higher redshifts, \cite{Gupta:2009} found that the number density of
21\,cm absorbers in known \mbox{Mg\,{\sc ii}} absorbers appeared to
decrease with redshift above $z \sim 1$, consistent with a reduction
in the CNM fraction. We pursue this idea further by investigating
whether future wide-field 21\,cm surveys can be used to measure the
average spin temperature in distant galaxies that are rich in atomic
gas.
\section{The expected number of intervening \mbox{H\,{\sc i}}
absorbers}\label{section:expected_number}
We estimate the expected number of intervening \mbox{H\,{\sc i}}
systems towards a sample of background radio sources by evaluating the
following integral over all sight-lines
\begin{equation}\label{equation:expected_number}
\mu = \iint{f(N_{\rm HI},X)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}},
\end{equation}
where $f(N_{\rm HI}, X)$ is the frequency distribution as a function
of column density ($N_{\rm HI}$) and comoving path length ($X$). We
use the results of recent surveys for 21\,cm emission in nearby
galaxies (e.g. \citealt{Zwaan:2005}) and high-redshift Lyman-$\alpha$
absorption in the Sloan Digitial Sky Survey (SDSS;
e.g. \citealt*{Prochaska:2005}; \citealt{Noterdaeme:2009}), which show
that $f(N_{\rm HI}, X)$ can be parametrized by a gamma function of the
form
\begin{equation}
f(N_{\rm HI}, X) = \left({f_{\ast} \over N_{\ast}}\right)\left({N_{\rm HI} \over N_{\ast}}\right)^{-\beta}\exp{\left(-{N_{\rm HI} \over N_{\ast}}\right)}\,\mathrm{cm}^{2},
\end{equation}
where $f_{\ast} = 0.0193$, $\log_{10}(N_{\ast}) = 21.2$ and $\beta =
1.24$ at $z = 0$ (\citealt{Zwaan:2005}), and $f_{\ast} = 0.0324$,
$\log_{10}(N_{\ast}) = 21.26$ and $\beta = 1.27$ at $z \approx 3$
(\citealt{Noterdaeme:2009}). While the observational data do not yet
constrain models for evolution of the \mbox{H\,{\sc i}} distribution
at intermediate redshifts between $z \sim 0.1$ and
$3$\footnote{Measurements of $f(N_{\rm HI},X)$ at intermediate
redshifts come from targeted ultra-violet surveys of DLAs using the
\emph{Hubble Space Telescope} (\citealt*{Rao:2006};
\citealt{Neeleman:2016}). However, due to the limited sample sizes
these are currently an order-of-magnitude less sensitive than the
nearby 21-cm and high-redshift optical Lyman-$\alpha$ surveys.}, it
is known to be much weaker than the significant decline seen in the
global star-formation rate and molecular gas over the same epoch
(e.g. \citealt{Lagos:2014}). We therefore carry out a simple linear
interpolation between the low and high redshift epochs to estimate
$f(N_{\rm HI},X)$ as a function of redshift.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{width_dist.pdf}
\caption{The distribution of 21\,cm line widths based on existing
detections of intervening absorption at $z > 0.1$ (see the text for
details of this sample). The sample size in each bin is denoted by
the number above and errorbars denote the standard deviation. The
solid red line is a log-normal fit to the data, from which we draw
random samples for our analysis.}\label{figure:width_dist}
\end{figure}
The probability of detecting an absorbing system of given column
density depends on the sensitivity of the survey, the flux density and
structure of the background source and the fraction of \mbox{H\,{\sc
i}} in the lower spin state, given by the spin temperature. We
express the column density ($N_{\rm HI}$; in atoms\,cm$^{-2}$) in
terms of the optical depth ($\tau$) and spin temperature ($T_{\rm
spin}$; in K) by
\begin{equation}\label{equation:column_density}
N_{\rm HI} = 1.823\times10^{18}\,T_{\rm spin} \int{\tau(v)\mathrm{d}v},
\end{equation}
where the integral is performed across the spectral line in the system
rest-frame velocity $v$ (in km\,s$^{-1}$). We then express the optical
depth in terms of the observables as
\begin{equation}
\tau = -\ln\left[1 + {\Delta{S}\over c_{\rm f}S_{\rm cont}}\right],
\end{equation}
where $\Delta{S}$ is the observed change in flux density due to
absorption, $S_{\rm cont}$ is the background continuum flux density
and $c_{\rm f}$ is the (often unknown) fraction of background flux
density subtended by the intervening gas.
\begin{figure}
\centering
\includegraphics[width=0.465\textwidth]{covfact_dist.pdf}
\caption{The distribution of \mbox{H\,{\sc i}} covering factors from
the main sample of \citet{Kanekar:2014a}, which were estimated using
the fraction of total continuum flux density in the quasar core. The
sample size in each bin is denoted by the number above and errorbars
denote the standard deviation. The solid red line is the uniform
distribution, from which we draw random samples for our
analysis.}\label{figure:covfact_dist}
\end{figure}
We assume that a single intervening system can be described by a
Gaussian velocity distribution of full width at half maximum (FWHM)
dispersion ($\Delta{v_{\rm 50}}$) and peak optical depth ($\tau_{\rm
peak}$), so that \autoref{equation:column_density} can be re-written
as
\begin{equation}\label{equation:column_density_gaussian}
N_{\rm HI} = 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm peak}\,\Delta{v_{\rm 50}}.
\end{equation}
If we further assume that the rms spectral noise is Gaussian, with a
standard deviation $\sigma_{\rm chan}$ per independent channel
$\Delta{v_{\rm chan}}$, then the 5$\sigma$ column density detection
limit is given by
\begin{equation}
N_{5\sigma} \approx 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm 5\sigma}\,\Delta{v_{\rm conv}},
\end{equation}
where
\begin{equation}\label{equation:optical_depth_limit}
\tau_{5\sigma} \approx -\ln\left[1 - {5\,\sigma_{\rm chan}\over c_{\rm f}\,S_{\rm cont}}\sqrt{\Delta{v}_{\rm chan}\over \Delta{v_{\rm conv}}}\right],
\end{equation}
and $\Delta{v_{\rm conv}} \approx \sqrt{\Delta{v}_{\rm chan}^{2} +
\Delta{v}_{50}^{2}}$, which is the observed width of the line, given
by the convolution of the physical velocity distribution and the
spectral resolution of the telescope. We now redefine $\mu$ as the
expected number of intervening \mbox{H\,{\sc i}} detections in our
survey as a function of the column density sensitivity along each
sight-line where each comoving path element
$\delta{X}(z)$\footnote{For the purposes of this work we adopt a flat
$\Lambda$ cold dark matter cosmology with $H_{0}$ =
70\,km\,s$^{-1}$, $\Omega_\mathrm{M}$ = 0.3 and $\Omega_{\Lambda}$ =
0.7. } in the integral defined by
\autoref{equation:expected_number} is given by
\begin{equation}
\delta{X}(z)=
\begin{cases}
{\delta{z}\,(1+z)^{2}\over \sqrt{(1+z)^{2}(1+z\Omega_{\rm
M})-z(z+2)\Omega_{\rm \Lambda}}}, & \text{if}\ N_{\rm HI} \geq N_{5\sigma}, \\
0, & \text{otherwise}.
\end{cases}
\end{equation}
To calculate the column density sensitivity for each comoving element
we draw random samples for $\Delta{v}_{50}$ and $c_{\rm f}$ from
continuous prior distributions based on existing evidence. In the case
of $\Delta{v}_{50}$ we use a log-normal distribution obtained from a
simple least-squares fit to the sample distribution from previous
21-cm absorption surveys reported in the literature (see
\autoref{figure:width_dist})\footnote{References for the literature
sample of line widths shown in \autoref{figure:width_dist}:
\citet*{Briggs:2001}; \citet*{Carilli:1993};
\citet*{Chengalur:1999}; \citet{Chengalur:2000, Curran:2007b,
Davis:1978, Ellison:2012, Gupta:2009, Gupta:2012, Gupta:2013};
\citet{Kanekar:2001b,Kanekar:2003b}; \citet{Kanekar:2001c,
Kanekar:2006, Kanekar:2009a, Kanekar:2013, Kanekar:2014a};
\citet{Kanekar:2003a}, \citet*{Kanekar:2007};
\citet*{Kanekar:2014b}; \citet{ Lane:2001, Lovell:1996, York:2007,
Zwaan:2015}.}, assuming that this correctly describes the true
distribution for the population of DLAs. However, direct measurement
of the \mbox{H\,{\sc i}} covering factor is significantly more
difficult and so for the purposes of this work we draw random samples
assuming a uniform distribution between 0 and 1. In
\autoref{figure:covfact_dist}, we show a comparison between this
assumption and the sample distribution estimated by
\cite{Kanekar:2014a} from their main sample of 37 quasars. Kanekar et
al. used VLBI synthesis imaging to measure the fraction of total
quasar flux density contained within the core, which was then used as
a proxy for the covering factor. By carrying out a two-tailed
Kolmogorov-Smirnov (KS) test of the hypothesis that the Kanekar et
al. data are consistent with our assumed uniform distribution, we find
that this hypothesis is rejected at the 0.05 level, but not at the
0.01 level (this outcome is dominated by the paucity of quasars in the
sample with $c_{\rm f} \lesssim 0.2$). It is therefore possible that
the population distribution of \mbox{H\,{\sc i}} covering factors may
deviate somewhat from the uniform distribution assumed in this
work. We discuss the implications of this further in
\autoref{section:covering_factor}.
\section{A 21\,cm absorption survey with
ASKAP}\label{section:all_sky_survey}
We use the Australian Square Kilometre Array Pathfinder (ASKAP;
\citealt{Johnston:2007}) as a case study to demonstrate the expected
results from planned wide-field surveys for 21\,cm absorption
(e.g. the ASKAP First Large Absorption Survey in \mbox{H\,{\sc i}} --
Sadler et al., the MeerKAT Absorption Line Survey -- Gupta et al., and
the Search for HI absorption with AperTIF -- Morganti et al.). ASKAP
is currently undergoing commissioning. Proof-of-concept observations
with the Boolardy Engineering Test Array (\citealt{Hotan:2014}) have
already been used to successfully detect a new \mbox{H\,{\sc i}}
absorber associated with a probable young radio galaxy at $z = 0.44$
(\citealt{Allison:2015a}). Here we predict the outcome of a future
2\,h-per-pointing survey of the entire southern sky ($\delta \leq
+10\degr$) using the full 36-antenna ASKAP in a single 304\,MHz band
between 711.5 and 1015.5\,MHz, equivalent to \mbox{H\,{\sc i}}
redshifts between $z = 0.4$ and 1.0.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{nsources_flux.pdf}
\caption{The number of radio sources in our simulated southern sky
survey ($\delta \leq +10\degr$) estimated from existing catalogues
at $L$-band frequencies (see the text for details). The grey region
encloses the expected number across the 711.5 - 1015.5\,MHz ASKAP
frequency band, assuming a canonical spectral index of $\alpha =
-0.7$.}\label{figure:nsources_flux}
\end{figure}
Our expectations of the ASKAP performance are based on preliminary
measurements by \cite{Chippendale:2015} using the prototype Mark {\sc
II} phase array feed. We estimate the noise per spectral channel
using the radiometer equation
\begin{equation}
\sigma_{\rm chan} = {S_{\rm system} \over \sqrt{n_{\rm pol}\,n_{\rm ant}\,(n_{\rm ant} - 1)\,\Delta{t}_{\rm in}\,\Delta{\nu}_{\rm chan}}},
\end{equation}
where $S_{\rm system}$ is the system equivalent flux density, $n_{\rm
pol}$ is the number of polarizations, $n_{\rm ant}$ is the number of
antennas, $\Delta{t}_{\rm in}$ is the on-source integration time and
$\Delta{\nu}_{\rm chan}$ is the spectral resolution in frequency. The
sensitivity of the telescope in the 711.5 - 1015.5\,MHz band is
expected to vary between $S_{\rm system} \approx 3200$ and $2000$\,Jy,
with the largest change in sensitivity between 700 and 800\,MHz. ASKAP
has dual linear polarization feeds, 36 antennas and a fine filter bank
that produces 16\,416 independent channels across the full 304\,MHz
bandwidth, so the expected noise per 18.5\,kHz channel in a 2\,h
observation is approximately 5.5 - 3.5\,mJy\,beam$^{-1}$ across the
band. In the case of an actual survey, the true sensitivity will of
course be recorded in the spectral data as a function of redshift (see
e.g. \citealt{Allison:2015a}), but for the purposes of the simulated
survey presented in this work we split the band into several frequency
bins to capture the variation in sensitivity and velocity resolution
(which is in the range 7.8\,km\,s$^{-1}$ at 711.5\,MHz to
5.5\,km\,s$^{-1}$ at 1015.5\,MHz).
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{zdist.pdf}
\caption{The distribution of CENSORS sources (\citealt{Brookes:2008})
brighter than 10\,mJy beyond a given redshift $z$. The red line
denotes the cumulative distribution calculated from the parametric
model of \citet{deZotti:2010}.}\label{figure:zdist}
\end{figure}
In order to simulate a realistic survey of the southern sky we select
all radio sources south of $\delta = +10\degr$ from catalogues of the
National Radio Astronomy Observatory Very Large Array Sky Survey
(NVSS, $\nu = 1.4$\,GHz, $S_{\rm src} \gtrsim 2.5$\,mJy;
\citealt{Condon:1998}), the Sydney University Molonglo SkySurvey ($\nu
= 843$\,MHz, $S_{\rm src} \gtrsim 10$\,mJy; \citealt{Mauch:2003}) and
the second epoch Molonglo Galactic Plane Survey ($\nu = 843$\,MHz,
$S_{\rm src} \gtrsim 10$\,mJy; \citealt{Murphy:2007}). The source flux
densities, used to calculate the optical depth limit in
\autoref{equation:optical_depth_limit}, are estimated at the centre of
each frequency bin by extrapolating from the catalogue values and
assuming a canonical spectral index of $\alpha = -0.7$. In
\autoref{figure:nsources_flux}, we show the resulting cumulative
distribution of radio sources in our sample as a function of flux
density across the band.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{nsources_tau.pdf}
\caption{The number of sources in our simulated ASKAP survey with a
21\,cm opacity sensitivity greater than or equal to
$\tau_{5\sigma}$, as defined by
\autoref{equation:optical_depth_limit}. The grey region encloses
opacity sensitivities for the 711.5 - 1011.5\,MHz band. Random
samples for the line FWHM and covering factor were drawn from the
distributions shown in \autoref{figure:width_dist} and
\autoref{figure:covfact_dist}.}\label{figure:nsources_tau}
\end{figure}
For any given sight-line, the redshift interval over which absorption
may be detected is dependent upon the distance to the continuum
source. The lack of accurate spectroscopic redshift measurements for
most radio sources over the sky necessitates the use of a statistical
approach based on a model for the source redshift distribution. We
therefore apply a statistical weighting to each comoving path element
$\delta{X}(z)$ such that the expected number of absorber detections is
now given by
\begin{equation}\label{equation:weighted_sum_number}
\mu = \iint{f(N_{\rm HI},X)\,\mathcal{F}_{\rm src}(z^{\prime} \geq z)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}},
\end{equation}
where
\begin{equation}
\mathcal{F}_{\rm src}(z^{\prime} \geq z) = {\int_{z}^{\infty} \mathcal{N}_{\rm src}(z^{\prime})\mathrm{d}z^{\prime}\over\int_{0}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},
\end{equation}
and $\mathcal{N}_{\rm src}(z)$ is the number of radio sources as a
function of redshift. To estimate $\mathcal{N}_{\rm src}(z)$ we use
the Combined EIS-NVSS Survey Of Radio Sources (CENSORS;
\citealt{Brookes:2008}), which forms a complete sample of radio
sources brighter than 7.2\,mJy at 1.4\,GHz with spectroscopic
redshifts out to cosmological distances. In \autoref{figure:zdist} we
show the distribution of CENSORS sources brighter than 10\,mJy beyond
a given redshift $z$, and the corresponding analytical function
derived from the model fit of \citet{deZotti:2010}, given by
\begin{equation}
\mathcal{N}_{\rm src}(z) \approx 1.29 + 32.37z - 32.89z^{2} + 11.13z^{3} - 1.25z^{4},
\end{equation}
which we use in our analysis. For the redshifts spanned by our
simulated ASKAP survey, the fraction of background sources evolves
from 87\,per\,cent at $z = 0.4$ to 53\,per\,cent at $z = 1.0$.
We assume that this redshift distribution applies to any sight-line
irrespective of the continuum flux density. However, this assumption
is only true if the source population in the target sample evolves
such that the effect of distance is nullified by an increase in
luminosity. Given this criterion, and the sensitivity of our simulated
survey, we limit our sample to sources with flux densities between 10
and 1000\,mJy, which are dominated by the rapidly evolving population
of high-excitation radio galaxies and quasars
(e.g. \citealt{Jackson:1999, Best:2012, Best:2014, Pracy:2016}) and
for which the redshift distribution is known to be almost independent
of flux density (e.g. \citealt{Condon:1984, Condon:1998}). In
\autoref{figure:nsources_tau}, we show the number of sources from this
sub-sample as a function of opacity sensitivity [as defined by
\autoref{equation:optical_depth_limit}], drawing random samples of the
line FWHM and covering factor from the distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. There
are approximately 190\,000 sightlines with sufficient sensitivity to
detect absorption of optical depth greater than $\tau_{5\sigma}
\approx 1.0$ and 25\,000 sensitive to optical depths greater than
$\tau_{5\sigma} \approx 0.1$. Since this distribution converges at
optical depth sensitivities greater than $\tau_{5\sigma} \approx 5$,
the population of sources fainter than 10\,mJy, which are excluded
from our simulated ASKAP survey, would not significantly contribute to
further detections of absorption. Similarly, while sources brighter
than 1\,Jy are good probes of low-column-density \mbox{H\,{\sc i}}
gas, they do not constitute a sufficiently large enough population to
significantly affect the total number of absorber detections expected
in the survey and can also be safely excluded.
Based on these assumptions, we can estimate the number of absorbers we
would expect to detect in our survey with ASKAP as a function of spin
temperature. In \autoref{figure:ndetections_nhi}, we show the expected
detection yield as a cumulative function of column density. We show
results for two scenarios where the spin temperature is fixed at a
single value of either 100 or 1000\,K, and the line FWHM and covering
factors are drawn from the random distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. We find
that for both these cases the expected number of detections is not
sensitive to column densities below the DLA definition of $N_{\rm HI}
= 2\times 10^{20}$\,cm$^{-2}$. We also show in
\autoref{figure:ndetections_total} the expected total detection yield
(integrated over all \mbox{H\,{\sc i}} column densities) as a function
of a single spin temperature $T_\mathrm{spin}$ and line FWHM
$\Delta{v}_\mathrm{50}$. We find that for typical spin temperatures of
a few hundred kelvin (consistent with the typical fraction of CNM
observed in the local Universe) and a line FWHM of approximately
$20$\,km\,s$^{-1}$, a wide-field 21\,cm survey with ASKAP is expected
to yield $\sim 1000$ detections. However, even moderate evolution to a
higher spin temperature in the DLA population should see significant
reduction in the detection yield from this survey.
\section{Inferring the average spin
temperature}\label{section:spin_temp}
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{ndetections_nhi.pdf}
\caption{The expected number of absorber detections (as a cumulative
function of column density) in our simulated ASKAP survey. We show
two scenarios for a single spin temperature $T_{\rm spin} = 100$ and
$1000$\,K, where we have drawn random samples for the line FWHM and
covering factor from the distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. In
both cases we find that the expected number of detections is not
sensitive to column densities below $N_{\rm HI} = 2 \times
10^{20}$\,cm$^{-2}$, indicating that such a survey will only be
sensitive to DLA systems.}\label{figure:ndetections_nhi}
\end{figure}
We cannot directly measure the spin temperatures of individual systems
without additional data from either 21\,cm emission or Lyman-$\alpha$
absorption. However, from \autoref{figure:ndetections_total} it is
evident that the total number of absorbing systems expected to be
detected with a reasonably large 21\,cm survey is strongly dependent
on the assumed value for the spin temperature. Therefore, by comparing
the actual survey yield with that expected from the known
\mbox{H\,{\sc i}} distribution, we can infer the average spin
temperature of the atomic gas within the DLA population for a given
redshift interval.
Assuming that the total number of detections follows a Poisson
distribution, the probability of detecting $\mathcal{N}$ intervening
absorbing systems is given by
\begin{equation} p(\mathcal{N}|\overline{\mu}) =
{\overline{\mu}^{\mathcal{N}} \over \mathcal{N}!}
\mathrm{e}^{-\overline{\mu}},
\end{equation}
where $\overline{\mu}$ is the expected total number of detections
given by the integral
\begin{equation}
\overline{\mu} = \iiint{\mu(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\rho(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\mathrm{d}T_{\rm spin}\mathrm{d}\Delta{v}_{50}\mathrm{d}c_{\rm f}},
\end{equation}
and $\rho$ is the distribution of systems as a function of spin
temperature, line FWHM and covering factor. We assume that all three
of these variables are independent\footnote{In the case where thermal
broadening contributes significantly to the velocity dispersion, and
the spin temperature is dominated by collisional excitation, the
assumption that these are independent may no longer hold. However,
given that collisional excitation dominates in the CNM, where
$T_{\rm spin} \sim 100\,\mathrm{K}$, the velocity dispersion would
have to satisfy $\Delta{v}_{50} \ll 10$\,km\,s$^{-1}$ (c.f. the
distribution shown in \autoref{figure:width_dist}).} so that $\rho$
factorizes into functions of each. We then marginalise over the
covering factor and line width distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist} so that
the expression for $\overline{\mu}$ reduces to
\begin{equation}\label{equation:harmonic_spin_temp}
\overline{\mu} = \int{\mu(T_{\rm spin})\rho(T_{\rm spin})\mathrm{d}T_{\rm spin}} = \mu(\overline{T}_{\rm spin}),
\end{equation}
where $\overline{T}_{\rm spin}$ is the harmonic mean of the unknown
spin temperature distribution, weighted by column density. This is
analogous to the spin temperature inferred from the detection of
absorption in a single intervening galaxy averaged over several
gaseous components at different temperatures
(e.g. \citealt{Carilli:1996}).
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{ndetections_total.pdf}
\caption{The expected total number of detections in our simulated
ASKAP survey, as a function of a single spin temperature ($T_{\rm
spin}$) and line FWHM ($\Delta{v}_{50}$). The vertical dotted
lines enclose the velocity resolution across the observed frequency
band. We draw random samples for the covering factor from a uniform
distribution between 0 and 1 (as shown in
\autoref{figure:covfact_dist}). The contours are truncated at $\mu <
1$ for clarity.}\label{figure:ndetections_total}
\end{figure}
In the event of the survey yielding $\mathcal{N}$ detections, we can
calculate the posterior probability density of $\overline{T}_{\rm
spin}$ using the following relationship between conditional
probabilities
\begin{equation}
p(\overline{T}_{\rm spin}|\mathcal{N}) = {p(\mathcal{N}|\overline{T}_{\rm spin})p(\overline{T}_{\rm
spin})\over p(\mathcal{N})},
\end{equation}
where $p(\overline{T}_{\rm spin})$ is our prior probability density
for $\overline{T}_{\rm spin}$ and $p(\mathcal{N})$ is the marginal
probability of the number of detections, which can be treated as a
normalizing constant. The minimally informative Jeffreys prior for the
mean value $\mu$ of a Poisson distribution is $1/\sqrt{\mu}$
(\citealt{Jeffreys:1946})\footnote{A suitable alternative choice for
the prior is the standard scale-invariant form $1/\mu$
(e.g. \citealt{Jeffreys:1961, Novick:1965, Villegas:1977}). While we
find that our choice of non-informative prior has negligible effect
on the spin temperature posterior for the full \mbox{H\,{\sc i}}
absorption survey, as one would expect this choice becomes more
important for smaller surveys. For the early-science 1000\,deg$^{2}$
survey discussed in \autoref{section:tspin_results} we find that the
difference in these two priors produces a $\sim 2$ to 20\,per\,cent
effect in the posterior. However, in all cases considered this
change is smaller than the 68.3\,per\,cent credible interval spanned
by the posterior.}. From \autoref{equation:harmonic_spin_temp} it
therefore follows that a suitable form for the non-informative spin
temperature prior is $p(\overline{T}_{\rm spin}) =
1/\sqrt{\overline{\mu}}$, so that
\begin{equation}\label{equation:tspin_prob}
p(\overline{T}_{\rm spin}|\mathcal{N}) = C^{-1}\,{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}},
\end{equation}
where the distribution is normalised to unit total probability by
evaluating the integral
\begin{equation}
C = \int{{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}}}\,\mathrm{d}\overline{T}_{\rm spin}.
\end{equation}
The probabilistic relationship given by \autoref{equation:tspin_prob}
and the expected detection yield derived in
\autoref{section:all_sky_survey} can be used as a frame-work for
inferring the harmonic-mean spin temperature using the results of any
homogeneous 21-cm survey. We have assumed that we can accurately
distinguish intervening absorbing systems from those associated with
the host galaxy of the radio source. However, any 21-cm survey will be
accompanied by follow-up observations, at optical and sub-mm
wavelengths, which will aid identification. Furthermore, future
implementation of probabilistic techniques to either use photometric
redshift information or distinguish between line profiles should
provide further disambiguation.
Of course we have not yet accounted for any error in our estimate of
$\overline{\mu}$, which will increase our uncertainty in
$\overline{T}_\mathrm{spin}$. In the following section we discuss
these possible sources of error and their effect on the result.
\section{Sources of error}\label{section:errors}
Our estimate of the expected number of 21\,cm absorbers is dependent
upon several distributions describing the properties of the foreground
absorbing gas and the background source distribution. For future
large-scale 21\,cm surveys, the accuracy to which we can infer the
harmonic mean of the spin temperature distribution will eventually be
limited by the accuracy to which we can measure these other
distributions. In this section, we describe these errors and their
propagation through to the estimate of $\overline{T}_\mathrm{spin}$,
summarizing our results in \autoref{table:tspin_uncertainties}.
\subsection{The covering factor}\label{section:covering_factor}
\subsubsection{Deviation from a uniform distribution between 0 and 1}
The fraction $c_{\rm f}$ by which the foreground gas subtends the
background radiation source is difficult to measure directly and is
thereby a significant source of error for 21\,cm absorption
surveys. In this work, we have assumed a uniform distribution for
$c_{\rm f}$, taking random values between 0 and 1. In
\autoref{section:expected_number}, we tested this assumption by
comparing it with the distribution of flux density core fractions in a
sample of 37 quasars, used by \cite{Kanekar:2014a} as a proxy for the
covering factor. By carrying out a two-tailed KS test, we found some
evidence (at the 0.05 level) that this quasar sample was inconsistent
with our assumption of a uniform distribution between 0 and
1. Noticeably there seems to be an under-representation of quasars in
the Kanekar et al. sample with estimated $c_{\rm f} \lesssim 0.2$. In
the low optical depth limit, the detection rate is dependent on the
ratio of spin temperature to covering factor, in which case a
fractional deviation in $c_{\rm f}$ will propagate as an equal
fractional deviation in $\overline{T}_\mathrm{spin}$. Based on the
difference seen in the covering factor distribution of the Kanekar et
al. sample and the uniform distribution, we assume that the spin
temperature can deviate by as much as $\pm$10\,per\,cent.
\subsubsection{Evolution with redshift}
We also consider that the covering factor distribution may evolve with
redshift, which would mimic a perceived evolution in the average spin
temperature. Such an effect was proposed by \cite{Curran:2006b} and
\cite{Curran:2012b}, who claimed that the relative change in
angular-scale behaviour of absorbers and radio sources between low-
and high-redshift samples could explain the apparent evolution of the
spin temperature found by \cite{Kanekar:2003b}. To test for this
effect in their larger DLA sample, \cite{Kanekar:2014a} considered a
sub-sample at redshifts greater than $z = 1$, for which the relative
evolution of the absorber and source angular sizes should be
minimal. While the significance of their result was reduced by
removing the lower redshift absorbers from their sample, they still
found a difference at $3.5\,\sigma$ significance between spin
temperature distributions in the two DLA sub-samples separated by a
median redshift of $z = 2.683$.
Future surveys with ASKAP and the other SKA pathfinders will search
for \mbox{H\,{\sc i}} absorption at intermediate redshifts ($z \sim
1$), where the relative evolution of the absorber and source angular
sizes is expected to be more significant than for the higher redshift
DLA sample considered by \cite{Kanekar:2014a}. We therefore consider
the potential effect of this cosmological evolution on the inferred
value of $\overline{T}_\mathrm{spin}$. We approximate the covering
factor using the following model of \cite{Curran:2006b}
\begin{equation}\label{equation:covering_factor}
c_{f} \approx
\begin{cases}
\left({\theta_{\rm abs}\over \theta_{\rm src}}\right)^{2}, & \text{if}\ \theta_{\rm abs} < \theta_{\rm src}, \\
1, & \text{otherwise},
\end{cases}
\end{equation}
where $\theta_{\rm abs}$ and $\theta_{\rm src}$ are the angular sizes
of the absorber and background source, respectively. Under the
small-angle approximation $\theta_{\rm abs} \approx {d_{\rm
abs}/D_{\rm abs}}$ and $\theta_{\rm abs} \approx {d_{\rm
src}/D_{\rm src}}$, where $d_{\rm abs}$ and $D_{\rm abs}$ are the
linear size and angular diameter distance of the absorber, and
likewise $d_{\rm src}$ and $D_{\rm src}$ are the linear size and
angular diameter distance of the background source. Assuming that the
ratio $d_{\rm abs}/d_{\rm src}$ is randomly distributed and
independent of redshift, any evolution in the covering factor is
therefore dominated by relative changes in the angular diameter
distances. We calculate the expected angular diameter distance ratio
at a redshift $z$ by
\begin{equation}
\left\langle{D_{\rm abs}\over D_{\rm src}}\right\rangle_{z} = D_{\rm abs}(z){\int_{z}^{\infty} \mathcal{N}_{\rm src}(z^{\prime})D_{\rm src}(z^{\prime})^{-1}\mathrm{d}z^{\prime}\over\int_{z}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},
\end{equation}
which, for the source redshift distribution model given by
\cite{deZotti:2010}, evolves from 0.7 at $z = 0.4$ to 1.0 at $z = 1.0$
(see \autoref{figure:dang_ratio}). We note that this is consistent
with the behaviour measured by \cite{Curran:2012b} for the total
sample of DLAs observed at 21\,cm wavelengths. By applying this as a
correction to the otherwise uniformly distributed covering factor
(using \autoref{equation:covering_factor}), we find that the inferred
value of $\overline{T}_{\rm spin}$ systematically increases by
approximately 30 per\,cent.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{dang_ratio.pdf}
\caption{The expected redshift behaviour of $D_{\rm abs}/D_{\rm src}$
based on the \citet{deZotti:2010} model for the radio source redshift
distribution.}\label{figure:dang_ratio}
\end{figure}
\subsection{The $\bmath{N_{\rm HI}}$ frequency distribution}
\subsubsection{Uncertainty in the measurement of $f(N_{\rm HI},X)$}
We assume that $f(N_{\rm HI}, X)$ is relatively well understood as a
function of redshift by interpolating between model gamma functions
fitted to the distributions at $z = 0$ and $3$. However, these
distributions were measured from finite samples of galaxies, which of
course have associated uncertainties that need to be considered. In
the case of the data presented by \cite{Zwaan:2005} and
\cite{Noterdaeme:2009}, both have typical measurement uncertainties in
$f(N_{\rm HI}, X)$ of approximately 10\,per\,cent over the range of
column densities for which our simulated ASKAP survey is sensitive
(see \autoref{figure:ndetections_nhi}). This will propagate as a
10\,per\,cent fractional error in the expected number of absorber
detections, and contribute a similar percentage uncertainty in the
inferred average spin temperature.
\subsubsection{Correcting for 21\,cm self-absorption}
In the local Universe, \cite{Braun:2012} showed that self-absorption
from opaque \mbox{H\,{\sc i}} clouds identified in high-resolution
images of the Local Group galaxies M31, M33 and the Large Magellanic
Cloud may necessitate a correction to the local atomic mass density of
up to 30\,per\,cent. Although it is not yet clear whether this small
sample of Local Group galaxies is representative of the low-redshift
population, it is useful to understand how this effect might propagate
through to our average spin temperature measurement. We therefore
replace the gamma-function parametrization of the local $f(N_{\rm
HI})$ given by \cite{Zwaan:2005} with the non-parametric values
given in table\,2 of \cite{Braun:2012}, and recalculate
$\overline{T}_{\rm spin}$. For an all-sky survey with the full
36-antenna ASKAP we find that $\overline{T}_{\rm spin}$ increases by
$\sim$30 for 100 detections and $\sim$10\,per\,cent for 1000
detections. Note that the correction increases for low numbers of
detections, which are dominated by the highest column density systems.
\subsubsection{Dust obscuration bias in optically-selected DLAs}
At higher redshifts, it is possible that the number density of
optically-selected DLAs could be significantly underestimated as a
result of dust obscuration of the background quasar
(\citealt{Ostriker:1984}). This would cause a reduction in the
$f(\mbox{H\,{\sc i}}, X)$ measured from optical surveys, thereby
significantly underestimating the expected number of intervening
21\,cm absorbers at high redshifts. The issue is further compounded by
the expectation that the highest column density DLAs ($N_{\rm HI}
\gtrsim 10^{21}$\,cm$^{-2}$), for which future wide-field 21\,cm
surveys are most sensitive (see \autoref{figure:ndetections_nhi}), may
contain more dust than their less-dense counterparts.
This conclusion was supported by early analyses of the existing quasar
surveys at that time (e.g. \citealt{Fall:1993}), which indicated that
up to 70\,per\,cent of quasars could be missing from optical surveys
through the effect of dust obscuration, albeit with large
uncertainties. However, subsequent optical and infrared observations
of radio-selected quasars (e.g. \citealt{Ellison:2001};
\citealt*{Ellison:2005}; \citealt{Jorgenson:2006}), which are free of
the potential selection biases associated with these optical surveys,
found that the severity of this issue was substantially over-estimated
and that there was minimal evidence in support of a correlation
between the presence of DLAs and dust reddening. Furthermore, the
\mbox{H\,{\sc i}} column density frequency distribution measured by
\cite{Jorgenson:2006} was found to be consistent with the
optically-determined gamma-function parametrization of
\cite{Prochaska:2005}, with no evidence of DLA systems missing from
the SDSS sample at a sensitivity of $N_{\rm HI} \lesssim 5 \times
10^{21}$\,cm$^{-2}$. Although radio-selected surveys of quasars are
free of the selection biases associated with optical surveys, they do
typically suffer from smaller sample sizes and are therefore less
sensitive to the rarer DLAs with the highest column densities.
Another approach is to directly test whether optically-selected
quasars with intervening DLAs, selected from the SDSS sample, are
systematically more dust reddened than a control sample of non-DLA
quasars. Comparisons in the literature are based on several different
colour indicators, which include the spectral index
(e.g. \citealt{Murphy:2004,Murphy:2016}), spectral stacking
(e.g. \citealt{Frank:2010, Khare:2012}) and direct photometry
(e.g. \citealt*{Vladilo:2008}; \citealt{Fukugita:2015}). The current
status of these efforts is summarized by \citet{Murphy:2016}, showing
broad support for a missing DLA population at the level of
$\sim$5\,per\,cent but highlighting that tension still exists between
different dust measurements. No substantial evidence has yet been
found to support a correlation between the dust reddening and
\mbox{H\,{\sc i}} column density in these optically selected DLA
surveys (e.g. \citealt{Vladilo:2008, Khare:2012, Murphy:2016}).
In an attempt to reconcile the differences and myriad biases
associated with these techniques, \cite{Pontzen:2009} carried out a
statistically-robust meta-analysis of the available optical and radio
data, using a Bayesian parameter estimation approach to model the dust
as a function of column density and metallicity. They found that the
expected fraction of DLAs missing from optical surveys is
7\,per\,cent, with fewer than 28\,per\,cent missing at 3\,$\sigma$
confidence. Based on this body of work we therefore assume that
approximately 10\,per\,cent of DLAs are missing from the SDSS sample
of \cite{Noterdaeme:2009} and consider the affect on our estimate of
$\overline{T}_{\rm spin}$. We further assume that there is no
dependance on column density, an assumption which is supported by the
aforementioned observational data for the range of column densities to
which our 21\,cm survey is sensitive. We find that increasing the
high-redshift column density frequency distribution by 10\,per\,cent
introduces a systematic increase of approximately 3\,per\,cent in the
expected number of detections for the redshifts covered by our ASKAP
surveys. We note that this error will increase significantly for
21\,cm surveys at higher redshifts where the optically derived
$f(\mbox{H\,{\sc i}}, X)$ dominates the calculation of the expected
detection rate.
\subsection{The radio source background}
As described in \autoref{section:all_sky_survey}, we weight the
comoving path-length for each sight-line by a statistical redshift
distribution in order to account for evolution in the radio source
background. We use the parametric model of \cite{deZotti:2010}, which
is derived from fitting the measured redshifts of \cite{Brookes:2008}
for CENSORS sources brighter than 10\,mJy, and assume that this
applies to all sources in the range 10 - 1000\,mJy. In
\autoref{figure:zdist}, we show the cumulative distribution of sources
located behind a given redshift and the associated measurement
uncertainty given by the errorbars. For the intermediate redshifts
covered by the ASKAP survey, the fractional uncertainty in this
distribution increases from $\sigma_{\mathcal{F}_{\rm
src}}/\mathcal{F}_{\rm src} \approx 3.5$ to 8\,per\,cent between
$z = 0.4$ and 1.0, which propagates through to a similar fractional
uncertainty in $\overline{T}_{\rm spin}$. However, for higher
redshifts this fractional uncertainty increases rapidly at $z > 2$, to
more than 50\,per\,cent at $z = 3$, reflecting the paucity of optical
spectroscopic data for the high-redshift radio source
population. Understanding how the radio source population is
distributed at lower flux densities and at higher redshifts is
therefore a concern for the future 21\,cm absorption surveys
undertaken with the SKA mid- and low-frequency telescopes (see
\citealt{Kanekar:2004} and \citealt*{Morganti:2015} for reviews).
\begin{table}
\begin{threeparttable}
\caption{An account of errors in our estimate of $\overline{T}_{\rm
spin}$ due to the accuracy to which we can determine the
expected number of absorber
detections.}\label{table:tspin_uncertainties}
\begin{tabular}{l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l}
\hline
& Source of error & $\mathrm{err}(\overline{T}_{\rm spin})$ & Refs. \\
& & [per\,cent] & \\
\hline
Covering factor & Distribution uncertainty & $\pm10$ & $a$ \\
Covering factor & Systematic evolution & +30 & $a$, $b$\\
$f(N_{\rm HI}, X)$ & Measurement uncertainty & $\pm10$ & $c, d$\\
Low-$z$ $f(N_{\rm HI}, X)$ & Systematic self-absorption & $+(10-30)$ & $e$ \\
High-$z$ $f(N_{\rm HI}, X)$ & Systematic dust-obscuration & $+3$ & $f$, $g$ \\
$\mathcal{F}_{\rm src}(z^{\prime} \geq z)$ & Measurement uncertainty & $\pm 5$ & $h$, $i$ \\
\hline
\end{tabular}
\begin{tablenotes}
\item[] References: $^{a}${\citet{Kanekar:2014a}},
$^{b}${\citet{Curran:2012b}}, $^{c}${\citet{Zwaan:2005}},
$^{d}${\citet{Noterdaeme:2009}} , $^{e}${\citet{Braun:2012}},
$^{f}${\citet{Pontzen:2009}}, $^{g}${\citet{Murphy:2016}},
$^{h}${\citet{Brookes:2008}}, $^{i}${\citet{deZotti:2010}}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\section{Expected results for future 21-cm absorption
surveys}\label{section:tspin_results}
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{tspin_prob.pdf}
\caption{The posterior probability density of the average spin
temperature, as a function of absorber detection yield
($\mathcal{N}$). We show results for our simulated all-southern-sky
survey with 2-h per pointing using the full 36-antenna ASKAP (top
panel) and a smaller 1000\,deg$^{2}$ survey with 12-h per pointing
and 12 antennas of ASKAP (bottom panel). The dashed curves show the
cumulative effect of the systematic errors discussed in
\autoref{section:errors}. $\overline{\mathcal{F}}_{\rm CNM}$ is the
average CNM fraction assuming a simple two-phase neutral ISM with
$T_{\rm spin,CNM} = 100$\,K and $T_{\rm spin,WNM} = 1800$\,K
(\citealt{Liszt:2001}).}\label{figure:tspin_prob}
\end{figure}
In the top panel of \autoref{figure:tspin_prob} we show the results of
applying our method for inferring $\overline{T}_{\rm spin}$ to the
simulated all-southern-sky \mbox{H\,{\sc i}} absorption survey with
ASKAP described in \autoref{section:all_sky_survey}. We account for
the uncertainties in the expected detection rate $\overline{\mu}$,
discussed in \autoref{section:errors}, by using a Monte Carlo approach
and marginalizing over many realizations. A yield of 1000 absorbers
from such a survey would imply an average spin temperature of
$\overline{T}_\mathrm{spin} =
127^{+14}_{-14}\,(193^{+23}_{-23})$\,K\footnote{We give the
68.3\,per\,cent interval about the median value measured from the
posterior distributions shown in \autoref{figure:tspin_prob}.},
where values in parentheses denote the alternative posterior
probability resulting from the systematic errors discussed in
\autoref{section:errors}. This scenario would indicate that a large
fraction of the atomic gas in DLAs at these intermediate redshifts is
in the classical stable CNM phase. Conversely, a yield of only 100
detections would imply that $\overline{T}_\mathrm{spin} =
679^{+64}_{-65}\,(1184^{+116}_{-120})$\,K, indicating that less than
10\,per\,cent of the atomic gas is in the CNM and that the bulk of the
neutral gas in galaxies is significantly different at intermediate
redshifts compared with the local Universe.
We also consider the effect of reducing the sky area and array size,
which is relevant for planned early science surveys with ASKAP and
other SKA pathfinder telescopes. In the bottom panel of
\autoref{figure:tspin_prob}, we show the spin temperatures inferred
when observing a random 1000\,deg$^{2}$ field for 12\,h per pointing,
between $z_{\rm HI} = 0.4$ and $1.0$, using a 12-antenna version of
ASKAP. We find that detection yields of 30 and 3 from such a survey
would give inferred spin temperatures of $\overline{T}_{\rm spin}
=134^{+23}_{-27}\,(209^{+40}_{-47})$ and
$848^{+270}_{-430}\,(1535^{+513}_{-837})$\,K, respectively. The
significant reduction in telescope sensitivity and sky-area,
compensated by the increase in integration time per pointing planned
for early-science, results in a factor of 30 decrease in the expected
number of detections and therefore an increase in the sample variance
and uncertainty in $\overline{T}_{\rm spin}$. However, this result
demonstrates that we expect to be able to distinguish between the
limiting cases of CNM-rich or deficient DLA populations even during
the early-science phases of the SKA pathfinders. For example 30
detections with the early ASKAP survey rules out an average spin
temperature of 1000\,K at high probability.
\section{Conclusions}
We have demonstrated a statistical method for measuring the average
spin temperature of the neutral ISM in distant galaxies, using the
expected detection yields from future wide-field 21\,cm absorption
surveys. The spin temperature is a crucial property of the ISM that
can be used to determine the fraction of the cold ($T_{\rm k} \sim
100$\,K) and dense ($n \sim 100$\,cm$^{-2}$) atomic gas that provides
sites for the future formation of cold molecular gas clouds and star
formation. Recent 21\,cm surveys for \mbox{H\,{\sc i}} absorption in
\mbox{Mg\,{\sc ii}} absorbers and DLAs towards distant quasars have
yielded some evidence of an evolution in the average spin temperature
that might reveal a decrease in the fraction of cold dense atomic gas
at high redshift (e.g. \citealt{Gupta:2009, Kanekar:2014a}).
By combining recent specifications for ASKAP, with available
information for the population of background radio sources, we show
that strong statistical constraints (approximately $\pm10$\,per\,cent)
in the average spin temperature can be achieved by carrying out a
shallow 2-h per pointing survey of the southern sky between redshifts
of $z = 0.4$ and $1.0$. However, we find that the accuracy to which we
can measure the average spin temperature is ultimately limited by the
accuracy to which we can measure the distribution of the covering
factor, the $N_{\rm HI}$ frequency distribution function and the
evolution of the radio source population as a function of redshift. By
improving our understanding of these distributions we will be able to
leverage the order-of-magnitude increases in sensitivity and redshift
coverage of the future SKA telescope, allowing us to measure the
evolution of the average spin temperature to much higher redshifts.
\section*{Acknowledgements}
We thank Robert Allison, Elaine Sadler and Michael Pracy for useful
discussions, and the anonymous referee for providing comments that
helped improve this paper. JRA acknowledges support from a Bolton
Fellowship. We have made use of \texttt{Astropy}, a
community-developed core \texttt{Python} package for astronomy
(\citealt{Astropy:2013}); NASA's Astrophysics Data System
Bibliographic Services; and the VizieR catalogue access tool, operated
at CDS, Strasbourg, France.
\bibliographystyle{mnras}
| {'timestamp': '2016-08-09T02:04:29', 'yymm': '1607', 'arxiv_id': '1607.04828', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04828'} | arxiv |
\section{Introduction}
Given $\rho>0$, we consider the problem
\begin{equation}\label{eq:main_prob_U}
\begin{cases}
-\Delta U + \lambda U = |U|^{p-1}U & \text{in }\Omega,\smallskip\\
\int_\Omega U^2\,dx = \rho, \quad U=0 & \text{on }\partial\Omega,
\end{cases}
\end{equation}
where $\Omega\subset{\mathbb{R}}^N$ is a Lipschitz, bounded domain, $1<p<2^*-1$, $\rho>0$ is a fixed
parameter, and both $U\in H^1_0(\Omega)$ and $\lambda\in{\mathbb{R}}$ are unknown. More precisely,
we investigate conditions on $p$ and $\rho$ (and also $\Omega$) for the solvability of
the problem.
The main interest in \eqref{eq:main_prob_U} relies on the investigation of standing wave
solutions for the nonlinear Schr\"odinger equation
\[
i\frac{\partial \Phi}{\partial t}+\Delta \Phi+ |\Phi|^{p-1}\Phi=0,\qquad
(t,x)\in {\mathbb{R}}\times \Omega
\]
with Dirichlet boundary conditions on $\partial\Omega$. This equation appears in several
different physical models, both in the case $\Omega={\mathbb{R}}^N$ \cite{MR2002047}, and on bounded
domains \cite{MR1837207}. In particular, the latter case appears in nonlinear optics and in
the theory of Bose-Einstein condensation, also as a limiting case of the equation on ${\mathbb{R}}^N$
with confining potential. When searching for solutions having the
wave function $\Phi$ factorized as $\Phi(x,t)=e^{i\lambda t} U(x)$, one obtains
that the real valued function $U$ must solve
\begin{equation}\label{eq:NLS}
-\Delta U + \lambda U = |U|^{p-1}U ,\qquad U\in H^1_0(\Omega),
\end{equation}
and two points of view are available. The first possibility is to assign the chemical
potential $\lambda\in{\mathbb{R}}$, and search for solutions of \eqref{eq:NLS} as critical points of the
related action functional. The literature concerning this approach is huge and we do not even
make an attempt to summarize it here. On the contrary, we focus on the second possibility,
which consists in considering $\lambda$ as part of the unknown and prescribing the mass (or
charge) $\|U\|_{L^2(\Omega)}^2$ as a natural additional condition. Up to our knowledge,
the only previous paper dealing with this case, in bounded domains, is \cite{MR3318740},
which we describe below. The problem of searching for normalized solutions in ${\mathbb{R}}^N$,
with non-homogeneous nonlinearities, is more investigated \cite{MR3009665,MR1430506},
even though the methods used there can not be easily extended to bounded domains, where
dilations are not allowed. Very recently, also the case of partial confinement has been
considered \cite{BeBoJeVi_2016}.
Solutions of \eqref{eq:main_prob_U} can be identified with critical points of the
associated energy functional
\[
\mathcal{E}(U) = \frac12\int_\Omega|\nabla U|^2\,dx - \frac{1}{p+1} \int_\Omega|U|^{p+1}\,dx
\]
restricted to the mass constraint
\[
{\mathcal{M}}_\rho=\{U\in H_0^1(\Omega) : \|U\|_{L^2(\Omega)}=\rho\},
\]
with $\lambda$ playing the role of a Lagrange multiplier.
A cricial role in the discussion of the above problem is played by the Gagliardo-Nirenberg
inequality: for any $\Omega$ and for any $v\in H^1_0(\Omega)$,
\begin{equation}
\label{sobest}
\|v\|^{p+1}_{L^{p+1}(\Omega)} \leq C_{N,p} \| \nabla v \|_{L^2(\Omega)}^{N(p-1)/2}
\| v \|_{L^2(\Omega)} ^{(p+1)-N(p-1)/2},
\end{equation}
the equality holding only when $\Omega={\mathbb{R}}^N$ and $v=Z_{N,p}$, the positive solution of $-\Delta Z + Z = Z^{p}$ (which is unique up to
translations \cite{MR969899}).
Accordingly, the exponent $p$ can be classified in relation with the so called
\emph{$L^2$-critical exponent} $1+4/N$ (throughout all the paper, $p$ will be always
Sobolev-subcritical and its criticality will be understood in the $L^2$ sense).
Indeed we have that ${\mathcal{E}}$ is bounded below and coercive on ${\mathcal{M}}_\rho$ if and only if
either $p$ is subcritical, or it is critical and $\rho$ is sufficiently small.
The recent paper \cite{MR3318740} deals with problem \eqref{eq:main_prob_U} in the case of the spherical domain $\Omega = B_1$, when searching for positive solutions $U$.
In particular, it is shown that the solvability of \eqref{eq:main_prob_U} is strongly influenced by the exponent $p$, indeed:
\begin{itemize}
\item in the subcritical case $1<p<1+4/N$, \eqref{eq:main_prob_U} admits a unique positive
solution for every $\rho>0$;
\item if $p=1+4/N$ then \eqref{eq:main_prob_U} admits a unique
positive solution for
\[
0<\rho<\rho^*=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}=\|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)},
\]
and no positive solutions for $\rho\geq\rho^*$;
\item finally, in the supercritical regime $1+4/N<p<2^*-1$, \eqref{eq:main_prob_U} admits positive
solutions if and only if $0<\rho\leq\rho^*$ (the threshold $\rho^*$ depending on $p$), and such
solutions are at least two for $\rho<\rho^*$.
\end{itemize}
In this paper we carry on such analysis, dealing with a general domain $\Omega$ and with solutions which
are not necessarily positive.
More precisely, let us recall that for any $U$ solving \eqref{eq:main_prob_U} for some $\lambda$, it is well-defined the Morse index
\[
m(U) = \max\left\{k : \begin{array}{l}
\exists V\subset H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\
\displaystyle\int_\Omega |\nabla v|^2 + \lambda v^2 - p|U|^{p-1}v^2\,dx<0
\end{array}
\right\}\in{\mathbb{N}}.
\]
Then, if $\Omega=B_1$, it is well known that a solution $U$ of \eqref{eq:main_prob_U} is
positive if and only if $m(U)=1$. Under this perspective, the results in \cite{MR3318740}
can be read in terms of Morse index one--solutions, rather than positive ones: introducing the sets of admissible masses
\[
{\mathfrak{A}}_k ={\mathfrak{A}}_k(p,\Omega) := \left\{\rho>0 : \begin{array}{l} \eqref{eq:main_prob_U}
\text{ admits a solution $U$ (for some $\lambda$)}\\ \text{having Morse index }m(U)\leq k
\end{array} \right\},
\]
then \cite{MR3318740} implies that ${\mathfrak{A}}_1(p,B_1)$ is a bounded interval if
and only if $p$ is critical or supercritical, while ${\mathfrak{A}}_1(p,B_1)={\mathbb{R}}^+$ in the subcritical case.
On the contrary, when considering general domains and higher Morse index, the situation may become
much more complicated. We collect some examples in the following remark.
\begin{remark}\label{rem:specialdomains}
In the case of a symmetric domain, one can use any solution as a building block to construct other solutions with
a more complex behavior, obtaining the so-called necklace solitary waves. Such kind of solutions are constructed
in \cite{MR3426917}, even though in such paper the focus is on stability, rather than on normalization conditions.
For instance, by scaling argument, any Dirichlet solution of $-\Delta U + \lambda U = |U|^{p-1}U$ in a rectangle $R=\prod_{i=1}^N(a_i,b_i)$ can
be scaled to a solution of $-\Delta U + k^2\lambda U = |U|^{p-1}U$ in $R/k$, $k\in{\mathbb{N}}_+$, and then $k^N$ copies of it can be
juxtaposed, with alternating sign. In this way one obtains a new solution on $R$ having $k^{4/(p-1)}$ times the mass of the starting one,
and eventually solutions in $R$ with arbitrarily high mass (but with higher Morse index) can be constructed even in the critical
and supercritical case. An analogous construction can be performed in the disk, using solutions in circular sectors as building blocks,
even though in this case explicit bounds on the mass obtained are more delicate.
Also, instead of symmetric domains, singular perturbed ones can be considered, such as dumbbell domains
\cite{MR949628}: for instance, using \cite[Theorem 3.5]{MR2997381}, one can show that for any $k$, there exists a domain $\Omega$,
which is close in a suitable sense to the disjoint union of $k$ domains, such that \eqref{eq:main_prob_U} has a \emph{positive}
solution on $\Omega$ with Morse index $k$ and $\rho=\rho_k\to+\infty$ as $k\to+\infty$.
This kind of results justifies the choice of classifying the solutions in terms of their Morse index, rather
than in terms of their nodal properties.
\end{remark}
Motivated by the previous remark, the first question we address in this paper concerns the boundedness of ${\mathfrak{A}}_k$.
We provide the following complete classification.
\begin{theorem}\label{thm:bbd_index}
For every $\Omega\subset{\mathbb{R}}^N$ bounded $C^1$ domain, $k\ge1$, $1<p<2^*-1$,
\[
\sup{\mathfrak{A}}_k(p,\Omega) < +\infty
\qquad\iff\qquad
p \ge 1+\frac{4}{N}.
\]
\end{theorem}
The proof of such result, which is outlined in Section \ref{sec:blow-up}, is obtained
by a detailed blow-up analysis of sequences of solutions with bounded Morse index, via
suitable a priori pointwise estimates (see \cite{MR2063399}). In this respect, the
regularity assumption on $\partial\Omega$ simplifies the treatment of possible
concentration phenomena towards the boundary. The argument, which holds
for solutions which possibly change sign, is inspired by \cite{MR2825606}, where the
case of positive solutions is treated.
Once Theorem \ref{thm:bbd_index} is established, in case $p\geq 1 + 4/N$ two questions arise, namely:
\begin{enumerate}
\item is it possible to provide lower bounds for $\sup{\mathfrak{A}}_k$? Is it true that $\sup{\mathfrak{A}}_k$ is strictly increasing in $k$, or, at least, that $\sup{\mathfrak{A}}_k > \sup{\mathfrak{A}}_1$ for some $k$?
\item is \eqref{eq:main_prob_U} solvable for every $\rho\in(0,\sup{\mathfrak{A}}_k)$, or at least can we characterize some subinterval of solvability?
\end{enumerate}
It is clear that both issues can be addressed by characterizing
values of $\rho$ for which existence (and multiplicity) of solutions with
bounded Morse index can be guaranteed. To this aim, it can be useful to restate
problem \eqref{eq:main_prob_U} as
\begin{equation}\label{eq:main_prob_u}
\begin{cases}
-\Delta u + \lambda u = \mu|u|^{p-1}u & \text{in }\Omega,\\
\int_\Omega u^2\,dx = 1, \quad u=0 & \text{on }\partial\Omega,
\end{cases}
\qquad\text{where}\quad
\begin{cases}
U=\sqrt{\rho} u\\
\mu = \rho^{(p-1)/2},
\end{cases}
\end{equation}
where now $\mu>0$ is prescribed. Since
\begin{equation}
\label{Emu}
\text{both } \mathcal{E}_{\mu}(u) := \frac{1}{2}\int_{\Omega}|\nabla u|^2- \frac{\mu}{p+1}\int_{\Omega}| u|^{p+1}
\qquad
\text{and }{\mathcal{M}}={\mathcal{M}}_1=\{u : \|u\|_{L^2(\Omega)}=1\}
\end{equation}
are invariant under the ${\mathbb{Z}}_2$-action of the involution $u\mapsto -u$, solutions of
\eqref{eq:main_prob_u} can be found via min-max principles in the framework of index
theories (see e.g. \cite[Ch. II.5]{St_2008}). Notice that in the supercritical case
${\mathcal{E}}_\mu$ is not bounded from below on ${\mathcal{M}}$. Following \cite{MR3318740}, it can be
convenient to parameterize solutions to \eqref{eq:main_prob_u} with respect to the
$H^1_0$-norm, therefore we introduce the sets
\begin{equation}\label{eq:defBU}
\mathcal{B}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx<\alpha\right\},\quad\quad
\mathcal{U}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx=\alpha\right\}.
\end{equation}
Introducing the first Dirichlet eigenvalue of $-\Delta$ in $H^1_0(\Omega)$,
$\lambda_1(\Omega)$, we have that the sets above are non-empty whenever $\alpha>
\lambda_1(\Omega)$.
Since we are interested in critical points
having Morse index bounded from above, following \cite{MR968487,MR954951,MR991264} we
introduce the following notion of genus.
\begin{definition}\label{def:genus}
Let $A\subset H^1_0(\Omega)$ be a closed set, symmetric with respect to the origin (i.e.
$-A=A$). We define the \emph{genus} $\gamma$ of a $A$ as
\[
\gamma(A) := \sup\{m : \exists h \in C({\mathbb{S}}^{m-1};A),\, h(-u)=-h(u)\}.
\]
Furthermore, we define
\[
\Sigma_{\alpha}=\{A\subset \overline{\mathcal{B}}_\alpha: A\text{ is closed and }-A=A\},
\qquad
\Sigma^{(k)}_{\alpha}=\{A\in \Sigma_{\alpha} : \gamma(A)\ge k\},
\]
\end{definition}
We remark that this notion of genus is different from the
classical one of \emph{Krasnoselskii genus}, which is well suited for estimates
of the Morse index from below, rather than above. Nonetheless, $\gamma$ shares with
the Krasnoselskii genus most of the main properties of an index
\cite{MR0163310,MR0065910}. In particular, by the Borsuk-Ulam Theorem, any set $A$
homeomorphic to the sphere ${\mathbb{S}}^{m-1} := \partial B_1 \subset {\mathbb{R}}^m$ has genus
$\gamma(A) = m$. Furthermore, we show in Section \ref{sec:2const} that $\Sigma^{(k)}_{\alpha}$ is not empty,
provided $\alpha>\lambda_k(\Omega)$ (the $k$-th Dirichlet eigenvalue of $-\Delta$ in $H^1_0(\Omega)$).
Equipped with this notion of genus we provide two different variational principles
for solutions of \eqref{eq:main_prob_u} (and thus of \eqref{eq:main_prob_U}). The first
one is based on a variational problem with \emph{two constraints}, which was exploited as the
main tool in proving the results in \cite{MR3318740}.
\begin{theorem}\label{thm:genus_2constr}
Let $k\geq1$ and $\alpha>\lambda_{k}(\Omega)$. Then
\begin{equation}
\label{maxmin}
M_{\alpha,\,k}:= \sup_{A\in\Sigma^{(k)}_{\alpha}}\inf_{u\in A}\int_{\Omega}|u|^{p+1}
\end{equation}
is achieved on ${\mathcal{U}}_\alpha$, and there exists a critical point
$u_\alpha\in {\mathcal{M}}$ such
that, for some $\lambda_\alpha\in{\mathbb{R}}$ and $\mu_\alpha>0$,
\begin{equation}
\label{lagreq}
\int_\Omega|\nabla u_\alpha|^2 = \alpha\qquad\text{and}\qquad -\Delta u_\alpha+\lambda_\alpha\,u_\alpha=\mu_\alpha |u_\alpha|^{p-1}u_\alpha\quad \text{in }\Omega.
\end{equation}
\end{theorem}
As a matter of fact, the results in \cite{MR3318740} were obtained by a detailed analysis of the map $\alpha \mapsto \mu_\alpha$ in the case $k=1$, i.e.
when dealing with
\[
M_{\alpha,1} = \max\left\{\|u\|_{L^{p+1}}^{p+1} : \|u\|_{L^2}^2=1,\,\|\nabla u\|_{L^2}^2=\alpha \right\}.
\]
In the present paper we do not investigate the properties of the map
$\alpha \mapsto \mu_\alpha$ for general $k$, but we rather prefer to exploit the
characterization of $M_{\alpha,k}$ in connection with
a second variational principle, which deals with only
\emph{one constraint}.
\begin{theorem}\label{thm:genus_1constr}
Let $1+{N}/{4}\leq p<2^*-1$. There exists a sequence $(\hat \mu_k)_k$
(depending on $\Omega$ and $p$) such that, for every $k\geq 1$ and $0<\mu<\hat \mu_k$, the
value
\begin{equation}
\label{infsuplev}
c_k:= \inf_{A\in\Sigma^{(k)}_{\alpha}}
\sup_{A}{\mathcal{E}}_\mu,
\end{equation}
is achieved in $\mathcal{B}_\alpha$, for a suitable $\alpha>\lambda_{k}(\Omega)$. Furthermore there exists a critical point $u_\mu\in {\mathcal{M}}$ such
that, for some $\lambda_\mu\in{\mathbb{R}}$,
\[
-\Delta u_\mu+\lambda_\mu\,u_\mu=\mu |u_\mu|^{p-1}u_\mu\quad \text{in }\Omega,
\]
$\|\nabla u\|_{L^2}^2<\alpha$, and $m(u_\mu)\le k$.
\end{theorem}
\begin{remark}
Of course, if $p<1+4/N$, the above theorem holds with $\hat\mu_k=+\infty$ for every $k$.
\end{remark}
\begin{corollary}
Let $\hat \rho_k := \hat \mu_k^{2/(p-1)}$. Then
\[
(0,\hat\rho_k) \subset {\mathfrak{A}}_k.
\]
\end{corollary}
The link between Theorem \ref{thm:genus_2constr} and Theorem \ref{thm:genus_1constr} is that we can provide explicit estimates
of $\hat \mu_k$ (and hence of $\hat\rho_k$) in terms of the map $\alpha\mapsto M_{\alpha,k}$ (see Section \ref{sec:1const}).
We stress that the above
results hold for any Lipschitz $\Omega$. As a first consequence, this allows to extend the
existence result in \cite{MR3318740} to non-radial domains.
\begin{theorem}\label{thm:intro_GS}
For every $0<\rho<\hat\rho_1=\hat\rho_1(\Omega,p)$ problem \eqref{eq:main_prob_U} admits a solution which is a local minimum of
the energy ${\mathcal{E}}$ on ${\mathcal{M}}_\rho$. In particular, $U$ is positive, has Morse index one and the associated solitary wave is orbitally
stable.
Furthermore, for every Lipschitz $\Omega$,
\begin{itemize}
\item $\displaystyle 1<p<1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)
= +\infty$,
\item $\displaystyle p=1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)
\geq \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$,
\item $\displaystyle 1+\frac{4}{N}<p<2^*-1 \implies \hat\rho_1\left(\Omega,p\right)
\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$,
\end{itemize}
where the universal constant $D_{N,p}$ is explicitly written in terms of $N$ and $p$ in
Section \ref{sec:1const}.
\end{theorem}
\begin{remark}\label{rem:introGS}
Of course, in the subcritical and critical cases, $c_1$ is actually a global minimum.
Furthermore, the lower bound for the supercritical case agrees with that of the critical one
since, as shown in Section \ref{sec:1const}, $D_{N,1+4/N} = \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$
(and $\lambda_1(\Omega)$ is raised to the $0^{\text{th}}$-power). Notice that the estimate for the supercritical case is new also in the case $\Omega=B_1$.
\end{remark}
We observe that the exponent of $\lambda_1(\Omega)$ in the supercritical threshold is negative,
therefore such threshold decreases with the size of $\Omega$.
Once the first thresholds have been estimated, we turn to the higher ones: by exploiting
the relations between $M_{\alpha,k}$ and $c_k$, we can show that the thresholds obtained
for Morse index one--solutions in Theorem \ref{thm:intro_GS} can be increased, by
considering higher Morse index--solutions, at least for some exponent.
\begin{proposition}\label{thm:intro_3>1}
For every $\Omega$ and $1<p<2^*-1$,
\[
\hat\rho_3\left(\Omega,p\right) \geq 2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.
\]
\end{proposition}
\begin{remark}
In the critical case, the lower bound for $\hat\rho_3$ provided by Proposition \ref{thm:intro_3>1} is twice that for $\hat\rho_1$ obtained in Theorem \ref{thm:intro_GS}.
By continuity, the estimate for $\hat\rho_3$ is larger than that for $\hat\rho_1$ also when
$p$ is supercritical, but not too large. To quantify such assertion, we can use Yang's inequality
\cite{MR1894540,MR2262780}, which implies that for every $\Omega$ it holds
\[
\lambda_3(\Omega)\leq \left(1+\frac{N}{4}\right)2^{2/N} \lambda_1(\Omega).
\]
We deduce that $2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}
\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$ whenever
\[
p\leq 1+\frac{4}{N} + \frac{8}{N^2\log_2\left(1+\frac{4}{N}\right)}.
\]
In particular, the physically relevant case $N=3$, $p=3$ is covered. Furthermore, if
$N\geq 7$, the above condition holds for every $p<2^*-1$.
\end{remark}
Beyond existence results for
\eqref{eq:main_prob_U}, also multiplicity results can be achieved. A first general
consideration, with this respect, is that Theorem \ref{thm:genus_1constr} holds true
also when using the standard Krasnoselskii genus instead of $\gamma$; this allows to
obtain critical points having Morse index bounded from below (see
\cite{MR968487,MR954951,MR991264}), and therefore to obtain infinitely many solutions,
at least when $\rho$ is less than some threshold. More specifically, we can also
prove the existence of a second solution in the supercritical case, thus extending to
any $\Omega$ the multiplicity result obtained in \cite{MR3318740} for the ball.
Indeed, on the one
hand, in the supercritical case ${\mathcal{E}}_\mu$ is unbounded from below; on the other hand
the solution obtained in Theorem \ref{thm:genus_1constr}, for $k=1$, is a local minimum.
Thus the Mountain Pass Theorem \cite{MR0370183} applies on $\mathcal{M}$, and a second solution can be
found for $\mu<\hat\mu_1$, see Proposition \ref{mpcritlev} for further details (and also
Remark \ref{rem:further_crit_lev} for an analogous construction for $k\ge2$).
To conclude this introduction, let us mention that the explicit lower bounds obtained in
Theorem \ref{thm:intro_GS} can be easily applied in order to gain
much more information also in the case of special domains, as those considered in
Remark \ref{rem:specialdomains}. For instance, we can prove then following.
\begin{theorem}\label{pro:symm}
Let $\Omega=B$ be a ball in ${\mathbb{R}}^N$. Then
\[
p<1+\frac{4}{N-1}
\quad\implies\quad
\text{\eqref{eq:main_prob_U} admits a solution for every }\rho>0.
\]
An analogous result holds when $\Omega=R$ is a rectangle, without further restrictions on
$p<2^*-1$.
\end{theorem}
Therefore our starting problem in $\Omega=B$ can be solved for any mass value also in the critical and
supercritical regime, at least for $p$ smaller than this further critical exponent $1+4/(N-1) > 1+ 4/N$.
Of course, higher masses require higher Morse index--solutions. In particular, since by \cite{MR3318740}
we know that ${\mathfrak{A}}_1(B,1+4/N) = (0,\|Z_{N,p}\|_{L^2})$, we have that for
larger masses, even though no positive solution exists, nodal solutions with higher
Morse index can be obtained: in such cases \eqref{eq:main_prob_U} admits \emph{nodal ground
states with higher Morse index}.
The paper is structured as follows: in Section \ref{sec:blow-up} we perform a blow-up analysis of
solutions with bounded Morse index, in order to prove Theorem \ref{thm:bbd_index}; Section \ref{sec:2const}
is devoted to the analysis of the variational problem with two constraints \eqref{maxmin}
and to the proof of Theorem \ref{thm:genus_2constr};
that of Theorems \ref{thm:genus_1constr}, \ref{thm:intro_GS} and Proposition
\ref{thm:intro_3>1} is developed
in Section \ref{sec:1const}, by means of the variational problem with one constraint
\eqref{infsuplev}; finally, Section
\ref{sec:symm} contains the proof of Theorem \ref{pro:symm}.
\textbf{Notation.} We use the standard notation
$\{\varphi_k\}_{k\geq1}$ for a basis of eigenfunctions of the Dirichlet laplacian in $\Omega$,
orthogonal in $H^1_0(\Omega)$ and orthonormal in $L^2(\Omega)$. Such functions are ordered in such
a way that the corresponding eigenvalues $\lambda_k(\Omega)$ satisfy
\[
0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\dots,
\]
and $\varphi_1$ is chosen to be positive on $\Omega$. $C_{N,p}$ denotes the universal constant in
the Gagliardo-Nirenberg inequality \eqref{sobest}, which is achieved (uniquely, up to translations and dilations)
by the positive, radially symmetric function $Z_{N,p}\in H^1({\mathbb{R}}^N)$, with
\[
\|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}.
\]
Finally, $C$ denotes every (positive) constant we need not to specify, whose value may change also within the same formula.
\section{Blow-up analysis of solutions with bounded Morse index}\label{sec:blow-up}
Throughout this section we will deal with a sequence $\{(u_n,\mu_n,\lambda_n)\}_n \subset H^1_0(\Omega)\times{\mathbb{R}}^+\times{\mathbb{R}}$ satisfying
\begin{equation}\label{eq:auxiliary_n}
-\Delta u_n+\lambda_n u_n=\mu_n |u_n|^{p-1}u_n,\qquad\int_\Omega u_n^2\, dx=1,\qquad \int_\Omega |\nabla u_n|^2\, dx=:\alpha_n.
\end{equation}
To start with, we recall the following result (actually, in \cite{MR3318740}, the result is stated for positive solution, but the proof
does not require such assumption).
\begin{lemma}[{\cite[Lemma 2.5]{MR3318740}}]\label{lemma:case_alpha_n_bounded}
Take a sequence $\{(u_n,\mu_n,\lambda_n)\}_n$ as in \eqref{eq:auxiliary_n}. Then
\[
\{\alpha_n\}_n \text{ bounded}
\qquad\implies\qquad
\{\lambda_n\}_n,\,\{\mu_n\}_n\text{ bounded}.
\]
\end{lemma}
Next we turn to the study of sequences having arbitrarily large $H^1_0$-norm. In particular, we will focus on sequences of solutions having a common upper
bound on the Morse index
\[
m(u_n) = \max\left\{k : \begin{array}{l}
\exists V\subset H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\
\displaystyle\int_\Omega |\nabla v|^2 + \lambda_n v^2 - p\mu_n|u_n|^{p-1}v^2\,dx<0
\end{array}
\right\}.
\]
Throughout this section we will assume that
\begin{equation}\label{eq:mainass_secMorse}
\text{the sequence }\{(u_n,\mu_n,\lambda_n)\}_n\text{ satisfies \eqref{eq:auxiliary_n},
with }\alpha_n\to+\infty\text{ and }m(u_n)\leq \bar k,
\end{equation}
for some $\bar k\in{\mathbb{N}}$ not depending on $n$.
\begin{lemma}\label{lem:lambda_bdd_below}
Let \eqref{eq:mainass_secMorse} hold. Then
\(
\lambda_n \geq -\lambda_{\bar k}(\Omega).
\)
\end{lemma}
\begin{proof}
Assume, to the contrary, that for some $n$ it holds $\lambda_n < -\lambda_{\bar k}(\Omega)$. For any real $t_1,\dots t_{\bar k}$
we define
\[
\phi := \sum_{h=1}^{\bar k} t_h \varphi_h.
\]
By denoting $J_{\lambda,\mu}(u)={\mathcal{E}}_\mu(u)+\frac{\lambda}{2}\|u\|_{L^2}^2$, so that
Morse index properties can be written in terms of $J''_{\lambda,\mu}$, we have
\[
\begin{split}
J''_{\lambda_n,\mu_n}(u_n)[u_n,\phi] &= -(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi,\\
J''_{\lambda_n,\mu_n}(u_n)[\phi,\phi] &= \sum_{h=1}^{\bar k} t_h^2 \int_\Omega \bigl (|\nabla \varphi_h| + \lambda_n \varphi_h^2\bigr )\,dx - p\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx \\
&\leq \sum_{h=1}^{\bar k} t_h^2(\lambda_{h}(\Omega) + \lambda_n) - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx \leq - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx,
\end{split}
\]
where equality holds if and only if $t_1=\dots=t_{\bar k}=0$. As a consequence
\begin{multline*}
J''_{\lambda_n,\mu_n}(u_n)[t_0 u_n+ \phi, t_0 u_n + \phi] \leq -t_0^2 (p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n^2 \\
- 2t_0(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi \,dx - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx.
\end{multline*}
We deduce that $J''_{\lambda_n,\mu_n}(u_n)$ is negative definite on $\spann\{u_n, \varphi_1,\dots,\varphi_{\bar k}\}$, in contradiction with the bound on the Morse index (note that
$u_n$ cannot be a linear combination of a finite number of eigenfunctions, otherwise using the equations we would obtain that such eigenfunctions are linearly dependent).
\end{proof}
\begin{lemma}
\label{localblow}
Let \eqref{eq:mainass_secMorse} hold.
Then $\lambda_n\to +\infty$.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:lambda_bdd_below} we have that $\lambda_n$ is bounded below. As a consequence, we can use H\"{o}lder inequality with $\|u_n\|_{L^2}=1$ and \eqref{eq:auxiliary_n} to write
\[
\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}\ge \mu_n \,\|u_n\|^{p+1}_{L^{p+1}}=\alpha_n+\lambda_n
\rightarrow +\infty.
\]
Let us define
\begin{equation}
\label{equn}
U_n: =\mu_n^{\frac{1}{p-1}}\,u_n,
\quad\text{ so that }
-\Delta U_n+\lambda_n U_n=|U_n|^{p-1}U_n\quad \text{in}\,\,\Omega,\quad
U|_{\partial\Omega}=0.
\end{equation}
Pick $P_n\in\Omega$ such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}$ and set
\begin{equation}
\label{tildepsn}
\tilde\varepsilon_n: =|U_n(P_n)|^{-\frac{p-1}{2}}=\frac{1}{\sqrt{\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}}}\longrightarrow 0
\end{equation}
Hence, $|U_n(P_n)|\to+\infty$; moreover, as $P_n$ is a point of positive maximum or of negative minimum, we have
\begin{equation}
\nonumber
0\le \frac{-\Delta U_n(P_n)}{U_n(P_n)}=|U_n(P_n)|^{p-1}-\lambda_n\,.
\end{equation}
Thus $\lambda_n|U_n(P_n)|^{1-p}\le 1$, and since $\lambda_n$ is bounded from below, we conclude
\begin{equation}
\label{limtildelam}
\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\longrightarrow \tilde\lambda\in [0,1].
\end{equation}
Now, we are left to prove that $\tilde\lambda>0$. Let us define
\begin{equation}
\label{tildeVn}
\tilde V_n(y)=\tilde\varepsilon_n^{\frac{2}{p-1}}\, U_n(\tilde\varepsilon_n\,y+P_n),\quad\quad y\in \tilde\Omega_n
:=\big (\Omega-P_n\big )/\tilde\varepsilon_n,
\end{equation}
and let $d_n := d(P_n,\partial\Omega)$; we have, up to subsequences,
\[
\frac{\tilde\varepsilon_n}{d_n}\longrightarrow L\in [0,+\infty]
\qquad\text{and}\qquad
\tilde\Omega_n\rightarrow\left\{
\begin{array}{ll}
{\mathbb{R}}^n, & \text{if $L=0$;} \\
H, & \text{if $L>0$,}
\end{array}
\right.
\]
where $H$ is a half-space such that $0\in \overline H$ and $d(0,\partial H)=1/L$.
The function $\tilde V_n$ satisfies
\begin{equation}
\nonumber
\left\{
\begin{array}{ll}
-\Delta \tilde V_n+\lambda_n\,\tilde \varepsilon_n^2\,\tilde V_n=|\tilde V_n|^{p-1}\tilde V_n, & \hbox{in}\,\, \tilde\Omega_n;\\
|\tilde V_n|\le |\tilde V_n(0)|=1, & \hbox{in}\,\, \tilde\Omega_n;\\
\tilde V_n=0, & \hbox{on}\,\, \partial\tilde \Omega_n.
\end{array}
\right.
\end{equation}
From \eqref{tildepsn} and \eqref{limtildelam} we get $\tilde\varepsilon_n^2\,\lambda_n\rightarrow \tilde\lambda$; hence, by elliptic regularity and up to a further subsequence, $\tilde V_n\rightarrow \tilde V$ in $\mathcal{C}^1_{{\mathrm{loc}}}(\overline H)$ where $\tilde V$ solves
\begin{equation}
\label{limprob1}
\left\{
\begin{array}{ll}
-\Delta \tilde V+\tilde\lambda\,\tilde V=|\tilde V|^{p-1}\tilde V, & \hbox{in}\,\, H;\\
|\tilde V|\le |\tilde V(0)|=1, & \hbox{in}\,\, H;\\
\tilde V=0, & \hbox{on}\,\, \partial H.
\end{array}
\right.
\end{equation}
Since $\sup_n m(U_n)\leq \bar k$ (as a solution to \eqref{equn}), one can show as in Theorem $3.1$ of \cite{MR2825606} that $m(\tilde V)\leq \bar k$. In particular, $\tilde V$ is stable outside a compact set (see Definition $2.1$ in \cite{MR2825606}) so that, by Theorem $2.3$ and Remark $2.4$ of \cite{MR2825606}, we have
$$\tilde V(x)\rightarrow 0 \quad\quad \text{as} \quad\quad |x|\rightarrow +\infty.$$
Moreover, since $\tilde V$ is not trivial, we also have that $\tilde\lambda>0$. For, if $\tilde\lambda=0$ the function
$\tilde V$ would be a solution of the Lane-Emden equation
$-\Delta u=|u|^{p-1}u$ either in ${\mathbb{R}}^n$ or in $H$. In both cases, $\tilde V$ would contradict Theorems $2$ and $9$ of \cite{MR2322150}, being non trivial and stable outside a compact set. Thus, $\tilde\lambda >0$ and by \eqref{limtildelam} we conclude $\lambda_n\rightarrow +\infty$.
\end{proof}
\begin{remark}\label{rem4}
We stress that the scaling argument in Lemma \ref{localblow}, leading to the limit problem \eqref{limprob1} (with $\tilde\lambda>0$), can be repeated also near points of \emph{local} extremum.
More precisely, let $Q_n$ be such that $|U_n(Q_n)|\to +\infty$ and
$$
|U_n(Q_n)|=\max_{\Omega\cap B_{R_n\tilde\varepsilon_n}(Q_n)}U_n,
$$
for some $R_n\to +\infty$. Then the above procedure can be repeated by replacing $P_n$ with $Q_n$ in definition \eqref{tildepsn}.
\end{remark}
The local description of the asymptotic behaviour of the solutions $U_n$ to \eqref{equn} with bounded Morse index can be carried out more conveniently by defining
the sequence (see \cite[Theorem $3.1$]{MR2825606})
\begin{equation}
\label{defVn}
V_n(y)=\varepsilon_n^{\frac{2}{p-1}}\, U_n(\varepsilon_n\,y+P_n),\quad y\in \Omega_n
:=\frac{\Omega-P_n}{\varepsilon_n},
\end{equation}
where $P_n$ is defined before \eqref{tildepsn}, and $\varepsilon_n=\frac{1}{\sqrt{\lambda_n}}\to 0$.
Then, $V_n$ satisfies
\begin{equation}
\nonumber
\left\{
\begin{array}{ll}
-\Delta V_n+ V_n=| V_n|^{p-1} V_n, & \hbox{in}\,\,\Omega_n;\\
|V_n|\le | V_n(0)|=\big ({\varepsilon_n/\tilde\varepsilon_n}\big )^{\frac{2}{p-1}}\rightarrow
\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, \Omega_n;\\
V_n=0, & \hbox{on}\,\, \partial\Omega_n.
\end{array}
\right.
\end{equation}
As before, we have (up to a subsequence) $V_n\rightarrow V$ in $\mathcal{C}^1_{\mathrm{loc}}(\overline H)$ where $H$ is either ${\mathbb{R}}^N$ or a half space and $V$ solves
\begin{equation}
\label{limprob2}
\left\{
\begin{array}{ll}
-\Delta V+ V=| V|^{p-1} V, & \hbox{in}\,\, H;\\
|V|\le | V(0)|=\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, H;\\
V=0, & \hbox{on}\,\, \partial H.
\end{array}
\right.
\end{equation}
By recalling the discussion following \eqref{limprob1} we also have $m(V)<+\infty$.
We collect some well known property of such a $V$ in the following result.
\begin{theorem}[\cite{MR688279,MR2825606,MR2322150,MR2785899}]\label{thm:unif_est_Farina}
Let $V$ be a classical solution to \eqref{limprob2} such that $m(V)\leq\bar k$. Then:
\begin{enumerate}
\item $H={\mathbb{R}}^N$;
\item $V(x)\to 0$ as $|x|\rightarrow +\infty$, $V \in H^1({\mathbb{R}}^N)\cap L^{p+1}({\mathbb{R}}^N)$;
\item there exist $C$ only depending on $\bar k$ (and not on $V$) such that
\[
\|V\|_{L^{\infty}} + \|\nabla V\|_{L^{\infty}}<C.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
Claim 2 follows from Theorem 2.3 and Remark 2.4 of \cite{MR2825606}, see also \cite[Remark 1.4]{MR688279}. As a consequence,
Theorem 1.1 of \cite[Remark 1.4]{MR688279} readily applies, providing claim 1 ($V$ is not trivial as $V(0)>0$). On the other hand, the $L^\infty$ estimates in claim 3.
are proved in Theorem 1.9 of \cite{MR2785899}.
\end{proof}
\begin{corollary}
\label{distpnbound}
If the sequence $\{U_n\}$ of solutions to \eqref{equn} has uniformly bounded Morse index, and if $P_n\in\Omega$ is such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}\to+\infty$, then
$$
\sqrt{\lambda_n}\,d(P_n,\partial\Omega)\rightarrow +\infty,\qquad\text{where }\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\to\tilde\lambda\in(0,1].
$$
\end{corollary}
\begin{remark}
Recall that $Z_{N,p}$, the unique positive solution to $-\Delta u+ u=| u|^{p-1} u$ in ${\mathbb{R}}^N$, has Morse index $1$ \cite{MR969899}; then, if $V$ solves \eqref{limprob2} in ${\mathbb{R}}^N$
and $1<m(V)<+\infty$, then $V$ is necessarily sign-changing.
\end{remark}
Following the same pattern as in \cite{MR2825606}, we now analyze the global behaviour of a sequence $\{U_n\}$ of solutions to \eqref{equn} for $\lambda_n\to +\infty$, assuming that
\(
\lim_{n\to +\infty} m(U_n)\leq\bar k<\infty.
\)
By the previous discussion, if $P^1_n$ is a sequence of points such that $|U_n(P^1_n)|=\|U_n\|_{L^{\infty}(\Omega)}$, we have
$|U_n(P^1_n)|\rightarrow +\infty$ and ${\lambda_n}\,d(P^1_n,\partial\Omega)^2\rightarrow +\infty$. We now look for other possible sequences of (local) extremum points $P^i_n$, $i=2,3,..$, along which $|U_n|$ goes to infinity. For any $R>0$, consider the quantity
\begin{equation}
\nonumber
h_1(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \Bigr ).
\end{equation}
We will prove that if $h_1(R)$ is \emph{not vanishing} for large $R$, then there exists a 'blow-up' sequence $P^2_n$ for $u_n$, 'disjoint' from $P^1_n$. Indeed, let us suppose that
\begin{equation}
\nonumber
\limsup_{R\to +\infty} h_1(R)=4\delta>0.
\end{equation}
Hence, up to a subsequence and for arbitrarily large $R$, we have
\begin{equation}
\label{ass1}
\lambda_n^{-\frac{1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \ge 2\delta.
\end{equation}
Since $U_n$ vanishes on $\partial\Omega$, there exists
$P^2_n\in\Omega\backslash B_{R\,\lambda_n^{-1/2}}(P_n^1)$
such that
\begin{equation}
\label{pn2}
|U_n(P_n^2)|=\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}|U_n(x)|.
\end{equation}
Clearly, assumption \eqref{ass1} implies that $|U_n(P_n^2)|\rightarrow +\infty$. We first prove that the sequences $P_n^1$ and $P_n^2$ are far away each other.
\begin{lemma}
\label{disj}
Take $R$ such that \eqref{ass1} holds, and let $P_n^2$ be defined as in \eqref{pn2}; then
\begin{equation}
\label{limp1p2}
\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow +\infty
\end{equation}
as $n\to \infty$.
\end{lemma}
\begin{proof}
Assuming the contrary one would get, up to a subsequence
\[
\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow R'\ge R.
\]
Let us now recall
that by \eqref{defVn} and the subsequent discussion, we have:
\begin{equation}
\label{limblowseq}
\lambda_n^{-\frac{1}{p-1}}\, U_n(\lambda_n^{-1/2}\,y+P^1_n) =: V^1_n(y)\rightarrow V(y)\quad \textrm{in}
\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)
\end{equation}
as $n\to +\infty$. Then, up to subsequences,
\begin{equation}
\nonumber
\lambda_n^{-\frac{1}{p-1}}\,|U_n(P_n^2)|=\big |V^1_n\bigr(\lambda_n^{1/2}(P_n^2-P^1_n)\bigl)\big|
\rightarrow \big |V(y')\big |,\quad |y'|=R'\ge R.
\end{equation}
Since $V$ is vanishing for $|y|\to +\infty$, one can choose $R$ such that
$|V(y)|\le\delta$ for every $ |y|\ge R$.
But this contradicts \eqref{ass1}.
\end{proof}
Furthermore, we also have that the blow-up points stay far away from the boundary.
\begin{lemma}
\label{distbd}
Assume \eqref{ass1} and let $P_n^2$ be defined as in \eqref{pn2}; then
\begin{equation}
\label{distp2nbound}
\sqrt{\lambda_n}\,d(P^2_n,\partial\Omega)\rightarrow +\infty
\end{equation}
as $n\to \infty$. Moreover,
\begin{equation}
\label{maxp2nball}
|U_n(P_n^2)|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^2_n)}|U_n|
\end{equation}
for some $R_n\to +\infty$.
\end{lemma}
\begin{proof}
Let us set
\begin{equation}
\nonumber
\tilde\varepsilon^2_n: =|U_n(P^2_n)|^{-\frac{p-1}{2}}\quad \mathrm{and}
\quad R_n^{(2)}:=\frac{1}{2}\,\frac{|P_n^2-P_n^1|}{\tilde\varepsilon^2_n}.
\end{equation}
Clearly, $\tilde\varepsilon^2_n\rightarrow 0$; moreover, by \eqref{ass1} and \eqref{pn2},
$\tilde\varepsilon^2_n\le (2\delta)^{-\frac{p-1}{2}}\lambda_n^{-1/2}$, so that
$$R^{(2)}_n\ge \frac{(2\delta)^{\frac{p-1}{2}}}{2}\,\lambda_n^{1/2}\,{|P_n^2-P_n^1|}
\rightarrow +\infty, $$
as $n\to +\infty$ by Lemma \ref{disj}. We claim that this implies
\begin{equation}
\label{maxp2n}
|U_n(P_n^2)|=\max_{\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)}| U_n|.
\end{equation}
For, if $x\in B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)$, by \eqref{limp1p2} we would have
$$|x-P^1_n|\ge |P^2_n-P^1_n|-|x-P^2_n|\ge \frac{1}{2}\,|P^2_n-P^1_n|\ge R\,\lambda_n^{-1/2},$$
for arbitrarily large $R$. This means that
$$\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)\subset \Omega\backslash B_{R\,\lambda_n^{-1/2}}(P^1_n).$$
Then, the claim follows. Now, by recalling Remark \ref{rem4}, we can apply to $U_n$ satisfying \eqref{maxp2n} the same scaling arguments as in the proof of Lemma \ref{localblow}, so that we conclude
$$
0< \lim_{n\to +\infty}\tilde\varepsilon_n^2\,\sqrt{\lambda_n}.
$$
Hence, \eqref{maxp2nball} holds by defining $R_n=R^{(2)}_n\tilde\varepsilon_n^2\,\sqrt{\lambda_n}$,
and \eqref{distp2nbound} follows by Corollary \ref{distpnbound}.
\end{proof}
We can now iterate the previous arguments: let us define, for $k\ge 1$,
\begin{equation}
\label{defhn}
h_k(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \Bigr ),
\end{equation}
where
$$d_{n,k}(x): =\min\{|x-P^i_n|\,:\, i=1,...,k\}$$
and the sequences $P^i_n$ are such that
\begin{equation}
\nonumber
\sqrt{\lambda_n}\,d(P^i_n,\partial\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j
\end{equation}
as $n\to +\infty$.
Assume that
$$\limsup_{n\to +\infty} h_k(R)=4\delta>0.$$
As before, up to a subsequence and for arbitrarily large $R$, we have
\begin{equation}
\label{assk}
\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \ge 2\delta
\end{equation}
and there exist $P^{k+1}_n$ so that
\begin{equation}
\nonumber
|U_n(P_n^{k+1})|=\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)|
\end{equation}
with $\lim_{n\to +\infty}|U_n(P_n^{k+1})|=+\infty$.
Moreover, as in Lemma \ref{disj} we deduce that, for every $i=1,...,k$
\begin{equation}
\label{limblowseqi}
\lambda_n^{-\frac{1}{p-1}}\,U_n(\lambda_n^{-1/2}\,y+P^i_n): = V^i_n(y)\rightarrow V^i(y)\quad \textrm{in}
\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)
\end{equation}
as $n\to +\infty$; hence, by \eqref{assk} and again from the vanishing of $V$ at infinity, we conclude that
\begin{equation}
\lambda_n^{1/2}|P_n^{k+1}-P^i_n|\rightarrow +\infty
\end{equation}
as $n\to \infty$, for every $i=1,...,k$. Setting now
\begin{equation}
\nonumber
\tilde\varepsilon^{k+1}_n: =|U_n(P^{k+1}_n)|^{-\frac{p-1}{2}}\quad \mathrm{and}
\quad R_n^{(k+1)}:=\frac{1}{2}\,\frac{d_{n,k}(P^{k+1}_n)}{\tilde\varepsilon^{k+1}_n}
\end{equation}
we still have $\tilde\varepsilon^{k+1}_n\to 0$ and, by \eqref{assk}, $R_n^{(k+1)} \to +\infty$ as $n\to \infty$ (see Lemma \ref{distbd}). Then, by the same arguments as in Lemma \ref{distbd}, we get
\begin{equation}
\label{maxpkn}
|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R^{(k+1)}_n\tilde\varepsilon^{k+1}_n}(P^{k+1}_n)} |u_n|\,,
\end{equation}
and furthermore
$$ \lim_{n\to +\infty}\tilde\varepsilon_n^{k+1}\,\sqrt{\lambda_n}>0\,,$$
so that by defining $R_n=:R^{(k+1)}_n\tilde\varepsilon_n^{k+1}\,\sqrt{\lambda_n}\rightarrow +\infty$
we have
\begin{equation}
\label{maxpknball}
|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{k+1}_n)}| U_n|.
\end{equation}
Now, by the same arguments as in \cite{MR2825606}, it turns out that the iterative procedure must stop after \emph{at most} $\bar k-1$ steps, where
$\bar k =\lim_{n\to +\infty} m(u_n)$. Thus, we have proved:
\begin{proposition}
\label{glob1}
Let $\{U_n\}_n$ be a solution sequence to \eqref{equn} such that $\lambda_n\to+\infty$
and $m(U_n)\leq\bar k$.
Then, up to a subsequence, there exist $P_n^1,...,P_n^k$, with $k\le \bar k$ such that
\begin{equation}
\label{limpin}
\sqrt{\lambda_n}\,d(P^i_n,\partial\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j
\end{equation}
as $n\to +\infty$ and
\begin{equation}
\nonumber
|U_n(P_n^{i})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{i}_n)}|U_n|,\quad i=1,...,k,
\end{equation}
for some $R_n\to +\infty$ as $n\to +\infty$. Finally,
\begin{equation}
\label{limhr0}
\lim_{R\to +\infty} h_k(R)=0
\end{equation}
where $h_k(R)$ is given by \eqref{defhn}.
\end{proposition}
We now show that the sequence $U_n$ decays exponentially away from the blow-up points.
\begin{proposition}
\label{glob2}
Let $\{U_n\}_n$ satisfy the assumptions of Proposition \ref{glob1}. Then, there exist
$P_n^1,...,P_n^k$ and
positive constants $C$, $\gamma$, such that
\begin{equation}
\label{stimglob}
|U_n(x)|\le C\lambda^{\frac{1}{p-1}}_n \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.
\end{equation}
\end{proposition}
\begin{proof}
By \eqref{limhr0}, for large $R>0$ and $n>n_0(R)$ it holds
\begin{equation}
\nonumber
\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \le \Bigr (\frac{1}{2p} \Bigl )^{\frac{1}{p-1} }
\end{equation}
Then, for $n>n_0(R)$ and for $x\in \{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$, we have
\begin{equation}
\nonumber
a_n(x): = \lambda_n-p |U_n(x)|^{p-1}\ge \lambda_n-\frac{\lambda_n}{2}=\frac{\lambda_n}{2}
\end{equation}
We stress that the linear operator
\begin{equation}
\nonumber
L_n: =-\Delta + a_n(x)
\end{equation}
comes from the linearization of equation \eqref{equn} at $U_n$; let us compute this operator on the functions
$$\phi^i_n(x)=e^{-\gamma\sqrt{\lambda_n}\,|x-P^i_n|}\,,\quad\quad \gamma>0,\quad\quad i=1,...,k$$
in $\{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$. We obtain:
$$L_n \phi^i_n(x)=\lambda_n\phi^i_n(x)\Bigr [-\gamma^2+(N-1)\frac{\gamma}{\sqrt{\lambda_n}\,|x-P^i_n|}
+\frac{a_n(x)}{\lambda_n}\Bigl ]\ge \lambda_n\phi^i_n(x)\bigr [-\gamma^2+1/2\bigl ]\ge 0$$
for $n$ large, provided $0<\gamma\le 1/\sqrt 2$. Moreover, for $|x-P^i_n|=R\lambda_n^{-1/2}$, $ i=1,...,k,$ and $R$ large we have
$$
e^{\gamma R}\phi^i_n(x)-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|=
1-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|>0
$$
as $n\to +\infty$, by \eqref{limblowseq}. Note further that
$$\{x:d_{n,k}(x)= R\,\lambda_n^{-1/2}\} = \bigcup_{i=1}^k \partial B_{R\,\lambda_n^{-1/2}}(P_n^i)
\subset \Omega$$
for large enough $n$. Then, by defining
\begin{equation}
\nonumber
\phi_n: = e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k \,\phi^i_n
\end{equation}
we have
$$\phi_n(x)-|U_n(x)|\ge 0\quad\quad \mathrm{on}\quad\quad\{d_{n,k}(x)= R\,\lambda_n^{-1/2}\}\cup\partial\Omega$$
and
\begin{equation}
\nonumber
L_n(\phi_n-|U_n|)\ge -L_n\,|U_n|=\Delta \,|U_n|-\lambda_n\,|U_n|+p|U_n|^p\ge (p-1)\,|U_n|^p\ge 0
\end{equation}
in $\Omega\backslash \{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$. Then (for $R$ large and $n\ge n_0(R)$)
we obtain $|U_n|\le \phi_n$ in the same set, by the minimum principle. Moreover, since by \eqref{limtildelam}
$$|U_n(x)|\le\|U_n\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le C \lambda_n^{\frac{1}{p-1}}$$
for some $C>0$, we also have, in $\{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$,
$$
|U_n(x)|\le\|U_n(x)\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le C e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k e^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}.
$$
Then, possibly by choosing a larger $C$, estimate \eqref{stimglob} follows for every $n$.
\end{proof}
We now exploit the previous results to show that suitable rescalings of the solutions to \eqref{eq:auxiliary_n} converge (locally) to some bounded solution $V$ of
\begin{equation}
\label{eqV}
-\Delta V+ V=| V|^{p-1} V
\end{equation}
in ${\mathbb{R}}^N$.
\begin{lemma}
\label{lemlim1}
Let \eqref{eq:mainass_secMorse} hold.
Then $|u_n|$ admits $k\le \bar k$ local maxima $P_n^1,...,P_n^k$ in $\Omega$ such that, defining
\begin{equation}
u_{i,n}(x)= \Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{1}{p-1}}u_n \bigr (\frac{x}
{\sqrt {\lambda_n}}+P_n^i\bigl ),\quad\quad x\in \Omega_{n,i}:=\sqrt{\lambda_n}\bigr (\Omega-P_n^i \bigl ),
\end{equation}
it results, up to a subsequence,
\begin{equation}
u_{i,n}(x)\rightarrow V_i\quad\quad \mathrm{in}\,\,\mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^n)\quad \mathrm{as}\,\,n\to +\infty,\quad \forall\,\,i=1,2,...,k,
\end{equation}
where $V_i$ is a bounded solution of \eqref{eqV} with $m(V_i)\le \bar{k}$.
\noindent As a consequence, for every $q\ge 1$,
\begin{equation}
\label{convlq}
\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx\rightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^q\,dx\quad\quad \mathrm{as}\,\, n\to +\infty.
\end{equation}
\end{lemma}
\begin{proof}
By Lemma \ref{localblow} we have $\lambda_n\to +\infty$; then, the first part of the lemma follows by definition \eqref{equn}, by \eqref{limblowseqi} and by Proposition \ref{glob1}; by the same proposition
and by Proposition \ref{glob2} we also have that the local maxima $P^i_n$ satisfies \eqref{limpin} and that the pointwise estimate
\begin{equation}
\label{stimglobvn}
|u_n(x)|\le C\Bigl (\frac{\lambda_n}{\mu_n}\Bigr )^{\frac{1}{p-1}} \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.
\end{equation}
holds. Let us fix $R>0$ and set $r_n=R/\sqrt{\lambda_n}$; for large enough $n$, \eqref{limpin} implies
$$B_{r_n}(P^i_n)\subset \Omega,\quad\quad B_{r_n}(P^i_n)\cap B_{r_n}(P^j_n)=\emptyset, \quad i\neq j.$$
Then we obtain
\[
\begin{array}{cl}
&\left |\left ( \frac{\mu_n}{\lambda_n}\right)^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{j=1}^{k}\int_{B_R(0)}|u_{j,n}|^q\,dx\,\,\right |
\smallskip\\&
=\left ( \frac{\mu_n}{\lambda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}\left |\int_{\Omega} |u_n|^q \,dx-
\sum_{j=1}^{k}\int_{B_{r_n}(P^j_n)}|u_{n}|^q\,dx\,\,\right |
\smallskip\\
&=\left ( \frac{\mu_n}{\lambda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} |u_n|^q \,dx
\le C^q\lambda_n^{N/2}
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} \left |\sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\right |^q \,dx
\smallskip\\
&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx
\smallskip\\
&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k
\int_{{\mathbb{R}}^N\backslash \,B_{r_n}(P^i_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx
\smallskip\\
&\le (Ck)^{q}\sum_{i=1}^k
\int_{{\mathbb{R}}^N\backslash \,B_{R}(0)} e^{-q\gamma\,|y|} \,dy\le C_1 \,e^{-C_2 R},
\end{array}
\]
for some positive $C_1$, $C_2$. Letting $n\to +\infty$ we have, up to subsequences,
\begin{multline*}
\Bigg |\lim_{n\to +\infty}\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{i=1}^{k}\int_{B_R(0)}|V_i|^q\,dx\,\,\Bigg |
\\
=\lim_{n\to +\infty}\Bigg |\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{i=1}^{k}\int_{B_R(0)}|u_{i,n}|^q\,dx\,\,\Bigg |\le C_1 \,e^{-C_2 R}.
\end{multline*}
Then, \eqref{convlq} follows by taking $R\to +\infty$.
\end{proof}
The previous lemma allows us to gain some information on the asymptotic behavior of the sequences $\lambda_n$, $\mu_n$ and $\|u_n\|_{L^{p+1}(\Omega)}$. We first provide some bounds for the solutions of the limit problem \eqref{eqV} which will be useful in the sequel.
\begin{lemma}
\label{boundbelow}
Let $V_i$, $i=1,\dots,k$ be as in Lemma \ref{lemlim1} (so that $m(V_i)\leq\bar k$). There exists a constant $C$, only depending on the full sequence $\{u_n\}_n$ and not on $V_i$
(and on the particular associated subsequence), such that
\[
\|V_i\|_{H^1}^2 = \|V_i\|_{L^{p+1}}^{p+1} \leq C.
\]
Furthermore, if also $m(V_i)\geq2$ (or, equivalently, if $V_i$ changes sign)
the following estimates hold:
\begin{equation}
\label{uppstiml2}
\|V_i\|^{p+1}_{L^{p+1}}> 2\,\|Z\|^{{p+1}}_{L^{{p+1}}},\qquad
\|V_i\|^2_{L^2}> 2\,\|Z\|^{2}_{L^{2}},
\end{equation}
where $Z\equiv Z_{N,p}$ is the unique positive solution to \eqref{eqV}.
\end{lemma}
\begin{proof}
To prove the bounds from above we claim that there exists $\bar R>0$, not depending on $i$, such that $V_i$ is stable outside $\overline{B_{\bar R}}$. Then
the desired estimate will follow, since
\[
\|V_i\|^{p+1}_{L^{p+1}} = \int_{B_{\bar R}} |V_i|^{p+1} + \int_{{\mathbb{R}}^N\setminus B_{\bar R}} |V_i|^{p+1},
\]
where the first term is uniformly bounded by Theorem \ref{thm:unif_est_Farina}, while the second one can be estimated in an uniform way
by reasoning as in the proof of \cite[Theorem 2.3]{MR2825606}. To prove the claim, recalling \eqref{defhn} and \eqref{limhr0}, let
$\bar R$ be such that
\[
h_k(\bar R) \leq \left(\frac{1}{p}\right)^{1/(p-1)}.
\]
Then $|V_i(x)|^{p-1}\leq 1/p $ on ${\mathbb{R}}^N\setminus B_{\bar R}$ and thus, for any $\psi\in C^\infty_0({\mathbb{R}}^N)$, $\psi\equiv0$ in $B_{\bar R}$, it holds
\[
\int_{{\mathbb{R}}^N} |\nabla \psi|^2 + \psi^2 - p|V_i|^{p-1}\psi^2\,dx \geq \left( 1 - p \|V_i\|^{p-1}_{L^{\infty}({\mathbb{R}}^N\setminus B_{\bar R})}\right)\int_{{\mathbb{R}}^N} \psi^2 \geq 0.
\]
Hence $V_i$ is stable outside $B_{\bar R}$, and the first part of the lemma follows.
On the other hand, if $V_i$ is a sign-changing solution to \eqref{eqV}, the associated energy functional
\begin{equation}
\nonumber
E(V_i)= \frac{1}{2}\|\nabla V_i\|^2_{L^2}+\frac{1}{2}\|V_i\|^2_{L^2}-\frac{1}{p+1}\|V_i\|^{p+1}_{L^{p+1}}
\end{equation}
satisfies the following \emph{energy doubling property} (see \cite{MR2263672}):
$$E(V_i)>2\,E(Z)$$
On the other hand, by using the equation $E'(V_i)V_i=0$ and the Pohozaev identity one gets
\begin{equation}
\label{eulp}
\|V_i\|^{p+1}_{L^{p+1}}= 2\,\frac{p+1}{p-1}\,E(V_i),\qquad
\|V_i\|^2_{L^2}= \frac{N+2-p\,(N-2)}{p-1}\,E(V_i)
\end{equation}
Since the ground state solution $Z$ satisfies the same identities, the bounds \eqref{uppstiml2} are readily verified.
\end{proof}
\begin{proposition}
Let \eqref{eq:mainass_secMorse} hold and the functions $V_i$ be defined as in Lemma
\ref{lemlim1}. We have, as $n\to +\infty$,
\begin{eqnarray}
\label{convl2}
{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-2/(p-1)}&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^2\,dx
\\
\label{convlp}
{\mu_n}^{\frac{p+1}{p-1}}\,\lambda_n^{N/2-(p+1)/(p-1)}\int_{\Omega} |u_n|^{p+1} \,dx&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx
\\
\label{convl2grad}
\alpha_n\,{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-(p+1)/(p-1)}&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx.
\end{eqnarray}
\end{proposition}
\begin{proof}
The limits \eqref{convl2} and \eqref{convlp} follow respectively by choosing $q=2$ and $q=p+1$ in \eqref{convlq} (recall that $\|u_n\|_{L^{2}}=1$). Furthermore, from the equations
for $u_n$ and $V_k$, we have
\[
\alpha_n+\lambda_n=\mu_n\|u_n\|_{L^{p+1}}^{p+1},\qquad
\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx + \int_{{\mathbb{R}}^n}|V_i|^2\,dx = \int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx,
\]
and also \eqref{convl2grad} follows.
\end{proof}
\begin{corollary}
\label{limmass}
With the same assumptions as above, we have that
\begin{enumerate}
\item if $1<p<1+\frac{4}{N}$, then $\mu_n\to +\infty$
\item if $p=1+\frac{4}{N}$, then $\mu_n\to \big (\sum_{i=1}^{k}\|V_i\|_{L^2}^2\big )^{2/N}\ge k^{2/N}
\|Z\|_{L^2}^{4/N}$
\item if $1+\frac{4}{N}<p<2^*-1$, then $\mu_n\to 0$.
\end{enumerate}
Furthermore
\begin{equation}
\label{limalphalam}
\frac{\alpha_n}{\lambda_n}\longrightarrow \frac{N(p-1)}{N+2-p(N-2)}.
\end{equation}
\end{corollary}
\begin{proof}
The limits of $\mu_n$ follow by the previous proposition. To prove the lower bound in $2$, recall that either $V_i=Z$ or $V_i$ satisfies \eqref{uppstiml2}. Finally, taking the quotient between \eqref{convl2grad} and \eqref{convl2}, we have
$$
\frac{\alpha_n}{\lambda_n}\longrightarrow
\frac{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx}{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}| V_i|^2\,dx}
$$
On the other hand, for every $i=1,2,...,k$ it holds
$$
\|\nabla V_i\|_{L^2}^2=\Bigg (\frac{\| V_i\|_{L^{p+1}}^{p+1}}{\|V_i\|_{L^2}^2} -1 \Bigg )\|V_i\|_{L^2}^2=
\frac{N(p-1)}{N+2-p(N-2)}\, \|V_i\|_{L^2}^2
$$
where the last equality follows by \eqref{eulp}. By inserting this into the above limit, we get \eqref{limalphalam}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:bbd_index}]
Let $(U_n,\lambda_n)$ solve \eqref{eq:main_prob_U}, with $\rho=\rho_n\to +\infty$
and $m(U_n)\leq k$. Changing variables as in \eqref{eq:main_prob_u}, we have that
$u_n=\rho_n^{-1/2}U_n$ satisfies \eqref{eq:auxiliary_n} with $\mu_n = \rho_n^{(p-1)/2} \to
+\infty$. As a consequence, Lemma \ref{lemma:case_alpha_n_bounded} guarantees that
$\alpha_n\to+\infty$, and Corollary \ref{limmass} yields $p<1+4/N$.
On the other hand, by direct minimization of the energy one can show that, if $p<1+4/N$, for
every $\rho>0$ there exists a solution of \eqref{eq:main_prob_U} having Morse index one (see
also Section \ref{sec:1const}).
\end{proof}
\begin{remark}
\label{limGN}
Reasoning as above we can also show that
\begin{equation}
\label{newcnp1}
\frac{\int_{\Omega} |u_n|^{p+1} \,dx}{\alpha_n^{N(p-1)/4}}\longrightarrow
C_{N,p}\,\frac{\|Z\|_{L^2}^{p-1}}{\big (\sum_{i=1}^{k}\| V_i\|_{L^2}^2\big )^{(p-1)/2}}.
\end{equation}
\end{remark}
\section{Max-min principles with two constraints}\label{sec:2const}
In this section we deal with the maximization problem with two constraints introduced in \cite{MR3318740}, aiming at considering more general max-min classes of critical points.
Let ${\mathcal{M}}$ be defined in \eqref{Emu} and, for any fixed $\alpha>\lambda_1(\Omega)$, let $\mathcal{B}_\alpha$, $\mathcal{U}_\alpha$ be defined as in \eqref{eq:defBU}. We will look for critical points of the $\mathcal{C}^2$ functional
\[
f(u)=\int_{\Omega}|u|^{p+1},\quad\quad\quad u\in {\mathcal{M}},
\]
constrained to $\mathcal{U}_\alpha$. To start with, we notice that the topological
properties of such set depend on $\alpha$.
\begin{lemma}\label{lemma:tilde_U_manifold}
Let $\alpha>\lambda_1(\Omega)$. Then the set
\[
{\mathcal{U}}_{\alpha}\setminus\left\{ \varphi\in {\mathcal{U}}_\alpha :
-\Delta\varphi = \alpha\varphi\right\}
\]
is a smooth submanifold of $H^1_0(\Omega)$ of codimension 2. In particular, this property
holds true for ${\mathcal{U}}_\alpha$ itself, provided $\alpha\neq\lambda_k(\Omega)$, for every $k$.
\end{lemma}
\begin{proof}
Let us set $F(u)=(\int_\Omega u^2\,dx-1, \ \int_\Omega|\nabla u|^2\,dx)$. For every
$u\in{\mathcal{U}}_\alpha$, if the range of $F'(u)$ is ${\mathbb{R}}^2$ then ${\mathcal{U}}_\alpha$ is a smooth manifold
at $u$. Since
\[
F'(u)[v]=2\left(\int_\Omega uv\,dx, \ \int_\Omega\nabla u\cdot\nabla v\,dx\right),
\qquad\text{for every }v\in H^1_0(\Omega),
\]
and $F'(u)[u]=2(1,\alpha)$, we have that $F'(u)$ is not surjective if and only if
\[
\int_\Omega\nabla u\cdot\nabla v\,dx = \alpha \int_\Omega uv\,dx
\qquad\text{for every }v\in H^1_0(\Omega). \qedhere
\]
\end{proof}
\begin{remark}
If $\varphi$ belongs to the eigenspace corresponding to $\lambda_k(\Omega)$, then
$\varphi \in {\mathcal{U}}_{\lambda_k(\Omega)}$. As a consequence ${\mathcal{U}}_{\lambda_k(\Omega)}$ may not be smooth
near $\varphi$. For instance, ${\mathcal{U}}_{\lambda_1(\Omega)}$ consists of two isolated points, $\pm\varphi_1$.
\end{remark}
Of course ${\mathcal{U}}_\alpha$ is closed and odd, for any $\alpha$. Recalling Definition
\ref{def:genus} we deduce that its genus
$\gamma({\mathcal{U}}_\alpha)$ is well defined.
\begin{lemma}
If $\alpha<\lambda_{k+1}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\leq k$.
\end{lemma}
\begin{proof}
Let $V_k:=\spann\{\varphi_1,\dots,\varphi_k\}$.
Since
\[
\min\left\{\int_\Omega |\nabla u|^2\,dx : u\in V_k^\perp,\,
\int_\Omega u^2\,dx=1\right\}=\lambda_{k+1}(\Omega),
\]
we have that ${\mathcal{U}} \cap V_k^\perp = \emptyset$, thus the projection
\[
g := \proj_{V_k} \colon {\mathcal{U}}_\alpha \to V_k\setminus\{0\}
\]
is a continuous odd map of ${\mathcal{U}}_\alpha$ into $V_k\setminus\{0\}$. Now, let $h\colon{\mathbb{S}}^{m}\to {\mathcal{U}}$ be continuous and odd.
Then $g\circ h$ is continuous and odd from ${\mathbb{S}}^{m}$ to $V_k\setminus\{0\}$, and Borsuk-Ulam's Theorem forces $m\leq k-1$.
\end{proof}
\begin{lemma}\label{lemma:genusbigger}
If $\alpha>\lambda_{k}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\geq k$.
\end{lemma}
\begin{proof}
To prove the lemma we will construct a continuous map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$. Let
$\ell\in{\mathbb{N}}$ be such that $\lambda_{\ell+1}(\Omega)>\alpha$. For every $i=1,\dots,k$ we define the
functions
\[
u_i:=\left(\frac{\lambda_{\ell+i}(\Omega)-\alpha}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_i
+\left(\frac{\alpha-\lambda_{i}(\Omega)}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_{\ell+i}.
\]
We obtain the following straightforward consequences:
\begin{enumerate}
\item as $\lambda_i(\Omega)<\alpha<\lambda_{\ell+i}(\Omega)$, for every $i$, $u_i$ is well defined;
\item $\int_\Omega u_i^2\,dx=1$, $\int_\Omega |\nabla u_i|^2\,dx=\alpha$;
\item for every $j\neq i$ it holds $\int_\Omega u_iu_j\,dx=\int_\Omega \nabla u_i\cdot\nabla u_j
\,dx=0$.
\end{enumerate}
Therefore the map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$ defined as
\[
h\colon x=(x_1,\dots,x_k) \mapsto \sum_{i=1}^k x_iu_i
\]
has the required properties.
\end{proof}
Now we turn to the properties of the functional $f$. To start with, it satisfies
the Palais-Smale (P.S. for short) condition on $\overline{\mathcal{B}}_{\alpha}$; more precisely, the following
holds.
\begin{lemma}
\label{psball}
Every P.S. sequence $u_n$ for $f\big |_{\overline{\mathcal{B}}_{\alpha}}$ is a P.S.
sequence for $f\big |_{\mathcal{U}_{\alpha}}$ and has a strongly convergent
subsequence in $\mathcal{U}_{\alpha}$.
\end{lemma}
\begin{proof}
We first show that there are no P.S. sequences in ${\mathcal{B}_{\alpha}}$. In fact, if $u_n$ is such a sequence, there is a sequence of real numbers $k_n$ such that
\begin{equation}
\label{ps}
\int_{\Omega}|u_n|^{p-1}u_n\,v-k_n\int_{\Omega}u_n\,v=o(1)\,\|v\|_{H^1_0}
\end{equation}
for every $v\in H^1_0(\Omega)$. Since $u_n$ is bounded in $H^1_0(\Omega)$, there is a subsequence (still denoted by $u_n$)
weakly convergent to $u\in H^1_0(\Omega)$; moreover, $u_n$ converges strongly in $L^{p+1}(\Omega)$ and in $L^2(\Omega)$ to the same limit. By choosing $v=u_n$, we see that $k_n$ is bounded, so that we can also assume that $k_n\rightarrow k$. By taking the limit of \eqref{ps} for $n\to\infty$ we get
\begin{equation}
\nonumber
\int_{\Omega}|u|^{p-1}u\,v=k\int_{\Omega}u\,v
\end{equation}
for every $v\in H^1_0(\Omega)$. Hence $u$ is constant, but this contradicts $u\in {\mathcal{M}}$.
Now, if $u_n$ is a P.S. sequence for $f$ on ${\mathcal{U}}_{\alpha}$, there are sequences of real numbers $k_n$, $l_n$ such that
\begin{equation}
\label{ps1}
\int_{\Omega}|u_n|^{p-1}u_n\,v-k_n\int_{\Omega}u_n\,v-l_n\int_{\Omega}\nabla u_n\,\nabla v=o(1)\,\|v\|_{H^1_0}.
\end{equation}
It is readily seen that $l_n$ is bounded away from zero, otherwise \eqref{ps1} is equivalent to \eqref{ps} (for some subsequence) and we still reach a contradiction. Then, we can divide both sides by $l_n$ and find that
there are sequences $\{\lambda_n\}_n$, $\{\mu_n\}_n$, with $\mu_n$ bounded, such that
\begin{equation}
\nonumber
\int_{\Omega}\nabla u_n\,\nabla v+\lambda_n\int_{\Omega}u_n\,v-\mu_n\int_{\Omega}|u_n|^{p-1}u_n\,v=o(1)\,\|v\|_{H^1_0}.
\end{equation}
Now, by reasoning as before one finds that
also the sequence $\{\lambda_n\}_n$ is bounded, so that by the relation
$$-\Delta u_n+\lambda_n u_n-\mu_n |u_n|^{p-1}u_n=o(1)\quad \mathrm{in}\,\, H^{-1}(\Omega)$$
and by the compactness of the embedding $H^1_0(\Omega)\hookrightarrow L^{p+1}(\Omega)$, the P.S. condition holds for the functional $f\big |_{{\mathcal{U}}_{\alpha}}$.
\end{proof}
We can combine the previous lemmas to prove one of the main results stated in the introduction.
\begin{proof}[Proof of Theorem \ref{thm:genus_2constr}]
Lemma \ref{psball} allows to apply standard variational methods (see e.g.
\cite[Thm. II.5.7]{St_2008}). We deduce that
$M_{\alpha,\,k}$ is achieved
at some critical point $u$ of $f\big |_{\mathcal{U}_{\alpha}}$.
This amounts to say that $u$ satisfies \eqref{lagreq}
for some real $\lambda$ and $\mu\neq 0$. We claim that there exists at least one
$u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ such that \eqref{lagreq} holds with $\mu>0$.
Assume by contradiction that for \emph{every} critical point of $f\big |_{\mathcal{U}_{\alpha}}$ at level $M_{\alpha,k}$ it holds $\mu< 0$ in equation \eqref{lagreq}.
Let us define the functional $T:\,H^1_0(\Omega)\to {\mathbb{R}}$ as
$$T(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2.$$
By denoting with $D$ the Fr\'{e}chet derivative and by $<\,,\,>$ the pairing between $H_0^1$ and its dual $H^{-1}$, our assumption can be restated as follows:
\noindent if there are $u\in f^{-1}(M_{\alpha,\,k})\cap\mathcal{U}_{\alpha}$ and $\mu\neq 0$ such that
\begin{equation}
\label{lagreq1}
\langle DT(u),\phi\rangle=\mu\langle Df(u),\phi\rangle
\end{equation}
for every $\phi\in H^1_0(\Omega)$ satisfying $\int_{\Omega}\phi u=0$ (that is for every $\phi$ tangent
to ${\mathcal{M}}$ at $u$) then $\mu<0$.
We stress that both $DT(u)$ and $Df(u)$ in the above equation are bounded away from zero, since there are no Dirichlet eigenfunctions in $\mathcal{U}_{\alpha}$ nor critical points of $f$ on ${\mathcal{M}}$.
Hence, by denoting with $\nabla_{T{\mathcal{M}}}$ the gradient of a functional (in $H^1_0$) in the direction tangent to ${\mathcal{M}}$, if $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ then $\nabla_{T{\mathcal{M}}}T(u)$ and
$\nabla_{T{\mathcal{M}}}f(u)$ \emph{are either opposite or not parallel}. Moreover, the angle between these (non vanishing) vectors is \emph{bounded away from zero}; otherwise, we would find sequences $u_n\in \mathcal{U}_{\alpha}$, $\mu_n>0$ such that
\begin{equation}
\label{noparal}
(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}-\mu_n(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=o(1)\|v\|_{H^1_0}
\end{equation}
for every $v\in H^1_0(\Omega)$; but since
$$(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}=\int_{\Omega}\nabla u_n\,\nabla v-\lambda_n^T\int_{\Omega}u_n\, v\,,$$
$$(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=\int_{\Omega}|u_n|^{p-1}u_n\,v-\lambda_n^f\int_{\Omega}u_n\, v\,,$$
for suitable bounded sequences $\lambda_n^T$, $\lambda_n^f$, this is equivalent to saying that $u_n$ is a P.S. sequence for
$f\big |_{\mathcal{U}_{\alpha}}$, so that, by Lemma \ref{psball}, we would get a constrained critical point with $\mu>0$.
Then, by choosing suitable linear combinations of the above tangential components
one can define a bounded $\mathcal{C}^1$ map $u\mapsto v(u)\in H_0^1(\Omega)$, with $v(u)$ tangent to ${\mathcal{M}}$
and satisfying the following property: there is $\delta>0$ such that
\begin{equation}
\label{diseqv}
\int_{\Omega}\nabla u\,\nabla v(u)< -\delta\, ,\quad\quad \int_{\Omega}|u|^{p-1}u\,v(u)>\delta\,,
\end{equation}
for every $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$. By continuity and possibly by decreasing $\delta$, inequalities \eqref{diseqv} extend to
\begin{equation}
\label{diseqv1}
f^{-1}(M_{\alpha,\,k}-\bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \big (\overline{\mathcal{B}}_{\alpha}
\backslash \overline{\mathcal{B}}_{\alpha-\tau}\big )
\end{equation}
for small enough, positive $\bar\varepsilon$ and $\tau$. Finally, since there are no critical points of $f$ in
${\mathcal{B}}_{\alpha}$ we can take that the \emph{second of \eqref{diseqv} holds on}
\begin{equation}
\label{diseqv2}
f^{-1}(M_{\alpha,\,k}- \bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \overline{\mathcal{B}}_{\alpha}.
\end{equation}
Let $\varphi$ be a $\mathcal{C}^1$ function on ${\mathbb{R}}$ such that:
$$0\le\varphi\le 1, \quad\varphi\equiv 1\,\, \mathrm{in}\,\,(M_{\alpha,\,k}-\bar\varepsilon/2, M_{\alpha,\,k}+\bar\varepsilon/2),\quad \varphi\equiv 0\,\, \mathrm{in}\,\,{\mathbb{R}}\backslash (M_{\alpha,\,k}-\bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon),$$
and define
\begin{equation}
\label{vectfield}
e(u)=\varphi(f(u))\,v(u).
\end{equation}
Clearly, $e$ is a $\mathcal{C}^1$ vector field on ${\mathcal{M}}$ and is uniformly bounded, so that there exists a global solution $\Phi(u,t)$ of the initial value problem
$$\partial_t\Phi(u,t)=e\big (\Phi(u,t)),\quad\quad \Phi(u,0)=0.$$
By definition \eqref{vectfield} and by the first of \eqref{diseqv} (on \eqref{diseqv1}) we get
$\Phi(u,t_0) \in \overline{\mathcal{B}}_{\alpha}$ for $t_0> 0$ and for any $u\in \overline{\mathcal{B}}_{\alpha}$; moreover, by the second inequality of \eqref{diseqv} (on \eqref{diseqv2}) there exists $\varepsilon\in (0,\bar\varepsilon)$ such that
$$f(\Phi(u,t_0))>M_{\alpha,\,k}+\varepsilon$$ for every $u\in f^{-1}(M_{\alpha,\,k}-\varepsilon, +\infty)\cap \overline{\mathcal{B}}_{\alpha}$.
Now, by \eqref{maxmin}, there is $A_{\varepsilon}\subset \overline{\mathcal{B}}_{\alpha}$ such that $\gamma(A_{\varepsilon})\ge k$ and
$$\inf_{u\in A_{\varepsilon}} f(u)\ge M_{\alpha,\,k}-\varepsilon.$$
Hence, $\gamma\big (\Phi(A_{\varepsilon},t_0) \big )\ge k$ and
$$\inf_{u\in \Phi(A_{\varepsilon},t_0)} f(u)\ge M_{\alpha,\,k}+\varepsilon$$
contradicting the definition of $M_{\alpha,\,k}$.
\end{proof}
\begin{remark}
\label{rem1}
If $\mu>0$, by testing \eqref{lagreq} with $u$ and by integration by parts we readily get
$\lambda>-\alpha$. An alternative lower bound, independent of $\alpha$, could be obtained
by adapting arguments from \cite{MR968487,MR954951,MR991264} in order to prove
that the Morse index of $u$ (as a solution of \eqref{lagreq}) is less or equal than $k$.
Then Lemma \ref{lem:lambda_bdd_below} would provide $\lambda\geq-\lambda_{k}$.
\end{remark}
\begin{remark}\label{rem:MvsCNp}
By the Gagliardo-Nirenberg inequality \eqref{sobest} we readily obtain that, for every $k\geq1$,
\[
M_{\alpha,k}\leq C_{N,p} \alpha^{N(p-1)/4}.
\]
Taking into account the previous remark, this agrees with Remark \ref{limGN}.
\end{remark}
We conclude this section with the following estimate.
\begin{lemma}
\label{lem:M3vsM1}
Under the assumptions and notation of Theorem \ref{thm:genus_2constr},
\[
M_{\alpha,3} \leq 2^{-(p-1)/2} M_{\alpha,1}.
\]
\end{lemma}
\begin{proof}
Let $A\in \Sigma^{(3)}_{\alpha}$, according to Definition \ref{def:genus}. Notice
that the map
\[
A\ni u \mapsto \left(\int_\Omega |u|u , \int_\Omega |u|^p u\right)\in{\mathbb{R}}^2
\]
is continuous and equivariant. By the Borsuk-Ulam Theorem, we deduce the existence of
$u_a\in A$ such that
\[
\int_\Omega |u^+_a|^2 = \int_\Omega |u^-_a|^2 = \frac12,\qquad
\int_\Omega |u^+_a|^{p+1} = \int_\Omega |u^-_a|^{p+1} =
\frac12 \int_\Omega |u_a|^{p+1},
\]
while
\[
\text{either }\int_\Omega |\nabla u^+_a|^2 \leq \frac{\alpha}{2}
\qquad
\text{or }\int_\Omega |\nabla u^-_a|^2 \leq \frac{\alpha}{2}.
\]
For concreteness let us assume that the first alternative holds;
as a consequence, we obtain that $v:=\sqrt2 u_a^+$ belongs to
$\overline{\mathcal{B}}_\alpha$. This yields
\[
M_{\alpha,1}\geq \int_\Omega |v|^{p+1} = 2^{(p+1)/2} \int_\Omega |u_a^+|^{p+1}
= \frac{2^{(p+1)/2}}{2} \int_\Omega |u_a|^{p+1} \geq 2^{(p-1)/2}
\inf_{u\in A} \int_\Omega |u|^{p+1},
\]
and since $A\in \Sigma^{(3)}_{\alpha}$ is arbitrary the proposition follows.
\end{proof}
\section{Min-max principles on the unit sphere in
\texorpdfstring{$L^2$}{L\texttwosuperior}}\label{sec:1const}
According to equation \eqref{Emu}, let ${\mathcal{M}}\subset H^1_0(\Omega)$ denote the unit
sphere with respect to the $L^2$ norm and $\mathcal{E}_{\mu}$ the energy functional
associated to \eqref{eq:main_prob_u}. In this section we are concerned with critical
points of $\mathcal{E}_{\mu}$ on ${\mathcal{M}}$ (which, in turn, correspond to solutions of our
starting problem \eqref{eq:main_prob_U}).
By the Gagliardo-Nirenberg inequality \eqref{sobest},
setting $\|\nabla u\|^2_{L^2}=\alpha$, one obtains
\begin{equation}\label{eq:boundonboundEmu}
\frac12\,\alpha- \mu\frac{C_{N,p}}{p+1}\,\alpha^{N(p-1)/4}
\leq \mathcal{E}_{\mu}(u)\le \frac12\alpha.
\end{equation}
In particular, $\mathcal{E}_{\mu}$ is bounded on any bounded subset of ${\mathcal{M}}$,
and it is bounded from below (and coercive) on the entire ${\mathcal{M}}$ for {subcritical} $p<1+4/N$
and for {critical} $p=1+4/N$ whenever $\mu< \frac{p+1}{2}C_{N,p}^{-1}$ .
In these cases, one can easily show that $\mathcal{E}_{\mu}$ satifies the P.S. condition
and apply the classical {minimax principle for even functionals} on a closed symmetric
submanifold (see e.g. \cite[Thm. II.5.7]{St_2008}).
In the complementary case, when $p$ is either supercritical, i.e. $p>1+4/N$, or
critical and $\mu$ is large, then $\mathcal{E}_{\mu}$ is not bounded from below (see e.g.
\eqref{minusinfty} below). In order to provide a minimax principle suitable for this case,
we recall the Definition \ref{def:genus} of genus and
that of $\mathcal{B}_\alpha$ (see equation \eqref{eq:defBU}). Furthermore, we denote with $K_{c}$ the (closed and symmetric) set of critical points of
${\mathcal{E}}_\mu$ at level $c$ contained in $\mathcal{B}_\alpha$. The following theorem is an adaptation of well known arguments
of previous critical point theorems relying on index theory.
\begin{theorem}
\label{infsupteo}
Let $k\ge1$, $\alpha>\lambda_k(\Omega)$, $\mu>0$ and $\tau>0$ be fixed, and let $c_k$ be defined as
in Theorem \ref{thm:genus_1constr}, equation \eqref{infsuplev}. If
\begin{equation}
\label{ass2}
c_k < \hat c_k:= \inf_{\substack{A\in\Sigma^{(k)}_{\alpha}\\
A\setminus\mathcal{B}_{\alpha-\tau}\neq\emptyset }}
\sup_{A\setminus\mathcal{B}_{\alpha-\tau}}{\mathcal{E}}_\mu,
\end{equation}
then $K_{c_k}\neq\emptyset$, and it contains a critical point of Morse index less or equal to $k$.
\end{theorem}
\begin{remark}
In case assumption \eqref{ass2} holds for $k,k+1,\dots,k+r$, and $c=c_k=...c_{k+r}$,
then it is standard to extend Theorem \ref{infsupteo} to obtain
\begin{equation}
\label{indexK}
\gamma(K_c)\ge r+1,
\end{equation}
so that $K_c$ contains infinitely many critical points.
\end{remark}
\begin{proof}[Proof of Theorem \ref{infsupteo}]
For any $a\in{\mathbb{R}}$ we denote by ${\mathcal{M}}_a$ the sublevel set $\{\mathcal{E}_\mu<a\}$.
First of all we notice that both $c_k$ and $\hat c_k$ are well defined and finite,
by Lemma \ref{lemma:genusbigger} and equation \eqref{eq:boundonboundEmu}.
Suppose now by contradiction that $K_{c_k}=\emptyset$. By a suitably modified version of
the Deformation Lemma (recall that ${\mathcal{E}}_\mu$ satisfies the P.S. condition on ${\mathcal{M}}$),
there exist $\delta>0$ and an equivariant homeomorphism
$\eta$ such that
$\eta(u)=u$ outside
$\mathcal{B}_{\alpha}\cap {\mathcal{M}}_{c_k+2\delta}$ and
\begin{equation}
\label{lowlev}
\eta({{\mathcal{M}}_{c_k+\delta}\cap \mathcal{B}_{\alpha-\tau}})\subset {\mathcal{M}}_{c_k-\delta}\cap
\mathcal{B}_{\alpha}.
\end{equation}
By definition of $c_{k}$ there exists $A\in \Sigma^{(k)}_{\alpha}$ such that
$A\subset {\mathcal{M}}_{c_k+\delta}$; it follows by assumption \eqref{ass2} (and by decreasing $
\delta$ if necessary) that $A\subset {\mathcal{M}}_{c_k+\delta}\cap \mathcal{B}_{\alpha-\tau}$.
Then, since $\eta$ is an odd homeomorphism, $\eta(A)\in \Sigma^{(k)}_{\alpha}$
and, by definition, $\sup_{\eta(A)}{\mathcal{E}}_\mu \ge c_k$, in contradiction with
\eqref{lowlev}. Finally, the estimate of the Morse index is a direct consequence of the
definition of genus we deal with: see \cite{MR968487}, Proposition on page 1030, or the discussion
at the end of Section 2 in \cite{MR991264}.
\end{proof}
We now provide a sufficient condition to guarantee the validity of assumption
\eqref{ass2}.
\begin{lemma}\label{lem:ckMak}
Let $k\ge1$, $\alpha>\lambda_k(\Omega)$ and $\mu>0$ satisfy
\begin{equation}
\label{muboundef}
0<\mu<\frac{p+1}{2}\,\frac{\alpha-\lambda_{k}(\Omega)}{M_{\alpha,k}-|\Omega|^{-\frac{p-1}{2}}}
\end{equation}
where $M_{\alpha,k}$ is defined in Theorem \ref{thm:genus_2constr}.
Then, for $\tau>0$ sufficiently small, \eqref{ass2} holds true.
\end{lemma}
\begin{proof}
We first estimate $c_k$ from above. To this aim, we construct a subset
$\tilde A\in \Sigma^{(k)}_{\alpha-\tau}$ (for any $\tau$ sufficiently small)
as
\begin{equation}
\label{Atilde}
\tilde A= \left\{\sum_{i=1}^k x_i\varphi_i : x=(x_1,\dots,x_k)\in{\mathbb{S}}^{k-1}\right\},
\end{equation}
where, as usual $\varphi_i$ denotes the Dirichlet eigenfunction associated to
$\lambda_i(\Omega)$. Indeed $\gamma(\tilde A)=k$ (it is homeomorphic to
${\mathbb{S}}^{k-1}$), and $\max_{u\in\tilde A}\|u\|^2_{H^1_0}=\lambda_{k}(\Omega)<\alpha-\tau$ for
$\tau$ small. Hence Holder inequality yields
\begin{equation}
\label{boundabove1}
c_k \leq \sup_{\tilde A}\mathcal{E}_{\mu}\le \frac{1}{2}\lambda_{k}(\Omega)
- \frac{\mu}{p+1}\,|\Omega|^{-\frac{p-1}{2}}.
\end{equation}
On the other hand, let $A\in\Sigma^{(k)}_{\alpha}$. Theorem \ref{thm:genus_2constr}
implies
\[
\inf_{u\in A} \int_\Omega |u|^{p+1} \leq M_{\alpha,k}.
\]
If moreover $A\setminus\mathcal{B}_{\alpha-\tau}\neq\emptyset$ we infer
\[
\sup_{A\setminus\mathcal{B}_{\alpha-\tau}}\mathcal{E}_{\mu}\ge
\frac{1}{2}(\alpha - \tau) - \frac{\mu}{p+1} M_{\alpha,k},
\]
and taking the infimum an analogous inequality holds true for $\hat c_k$. Comparing
with \eqref{boundabove1} the lemma follows.
\end{proof}
Exploiting the results above, we are ready to prove our main existence results.
\begin{proof}[End of the proof of Theorem \ref{thm:genus_1constr}]
By Theorem \ref{infsupteo} and Lemma \ref{lem:ckMak} the proof is completed by choosing
\[
\hat\mu_k:=\sup_{\alpha>\lambda_k(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{k}(\Omega)}{M_{\alpha,k}-|\Omega|^{-\frac{p-1}{2}}}.
\qedhere
\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:intro_GS}]
We write the proof in terms of ${\mathcal{E}}_\mu$, the theorem following by the relations in
\eqref{eq:main_prob_u}. Recall that, for every $u\in \overline{\mathcal{B}}_\alpha$, $\gamma
\left(\{u,-u\}\right)=1$. We deduce that $c_1$ is actually a local
minimum for ${\mathcal{E}}_\mu$, achieved by some $u$ which solves \eqref{eq:main_prob_u}
(for a suitable $\lambda$), and it can be chosen positive by symmetry.
Since
\[
\int_\Omega |\nabla u|^2 + \lambda u^2 - p\mu|u|^{p+1}\,dx=-(p-1)\int_\Omega \mu|u|^{p+1}\,dx<0,
\]
and $H^1_0(\Omega) = \spann\{u\}\oplus T_{u}\mathcal{M}$, we have that $u$
has Morse index $1$. In a standard way, the minimality
property of $u$ implies also orbital stability of the associated solitary wave (see e.g.
\cite{MR677997}). Turning to the estimates for $\hat\mu_1 = \hat\rho_1^{(p-1)/2}$,
we can deduce it using Lemma \ref{lem:ckMak} and Remark \ref{rem:MvsCNp}, which yield
\[
\hat\mu_1\left(\Omega,p\right):=\sup_{\alpha>\lambda_1(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{1}(\Omega)}{C_{N,p} \alpha^{\frac{N(p-1)}{4}}-|\Omega|^{-\frac{p-1}{2}}}
\geq \frac{p+1}{2C_{N,p}}\,\sup_{\alpha>\lambda_1(\Omega)}\frac{\alpha-\lambda_{1}
(\Omega)}{\alpha^{\beta}},
\]
where $\beta:=N(p-1)/4$. Now, if $\beta\leq1$ we obtain the desired bound for the
subcritical and critical cases. On the other hand, when $\beta>1$, elementary
calculations show that
\[
\hat\mu_1\left(\Omega,p\right)\geq \frac{p+1}{2C_{N,p}}
\,\frac{(\beta-1)^{(\beta-1)}}{\beta^\beta}\, \lambda_1(\Omega)^{-(\beta-1)},
\]
and finally
\[
\hat\rho_1\left(\Omega,p\right)\geq \underbrace{\left[\frac{p+1}{2C_{N,p}}
\,\frac{(\beta-1)^{(\beta-1)}}{\beta^\beta}\right]^{\frac{2}{p-1}}}_{D_{N,p}}\,
\lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.\qedhere
\]
\end{proof}
\begin{proof}[Proof of Proposition \ref{thm:intro_3>1}]
As usual, by \eqref{eq:main_prob_u}, we have to prove that
\[
\hat\mu_3\left(\Omega,p\right)\geq
2^{(p-1)/2} D_{N,p}\lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.
\]
By Lemmas \ref{lem:ckMak}, \ref{lem:M3vsM1}, and Remark \ref{rem:MvsCNp} we obtain
\[
\begin{split}
\hat\mu_3 &= \sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}(\Omega)}{M_{\alpha,3}-|\Omega|^{-\frac{p-1}{2}}} \geq
\sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}(\Omega)}{
2^{-(p-1)/2}M_{\alpha,1}-|\Omega|^{-\frac{p-1}{2}}}\\
&\geq 2^{(p-1)/2}\sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}
(\Omega)}{C_{N,p}\alpha^\beta-2^{(p-1)/2}|\Omega|^{-\frac{p-1}{2}}},
\end{split}
\]
where $\beta:=N(p-1)/4$, and the desired result follows by arguing as in the proof of
Theorem \ref{thm:intro_GS}.
\end{proof}
To conclude this section we prove that in the supercritical case, if $\mu$ is not too
large, in addition to $(c_k)_k$ there is a further sequence of critical levels
$(\bar c_k)$ of $\mathcal{E}_{\mu}$ constrained to $\mathcal{M}$. For concreteness,
let us first consider the case $k=1$: since in such case $c_1$ is a local minimum of
${\mathcal{E}}_\mu$ in $\mathcal{M}$, and ${\mathcal{E}}_\mu$ is unbounded from below in $\mathcal{M}$,
the critical level $\bar c_1$ is of mountain pass type.
\begin{proposition}
\label{mpcritlev}
Let $p>1+4/N$, $\mu<\hat\mu_1$, and $u_1$ denote the local minimum point of ${\mathcal{E}}_\mu$
in $\mathcal{M}$, according to Theorems \ref{infsupteo} and \ref{thm:intro_GS}. The value
\[
\bar c_1 : =\inf_{\gamma\in \Gamma}\sup_{[0,1]}\mathcal{E}_{\mu}(\gamma(s)),
\quad\text{where }\Gamma:=\left\{\gamma\in C([0,1];{\mathcal{M}}) : \gamma(0)=u_1,\,\gamma(1)<c_1-1\right\},
\]
is a critical level for ${\mathcal{E}}_\mu$ in $\mathcal{M}$.
\end{proposition}
\begin{proof}
Notice that, if $p>1+4/N$, then $\mathcal{E}_{\mu}\to -\infty$ along some sequence in
${\mathcal{M}}$. Indeed, by defining
\begin{equation}
\label{concfun}
w_n(x): = \eta(x)Z_{N,p}\big ((x-x_0)/a_n\big )\quad \text{and}\quad
\tilde{w_n}:=\frac{w_n}{\| w_n\|^2_{L^2(\Omega)}}\,\in\, {\mathcal{M}},
\end{equation}
where $a_n\to 0^+$, $x_0\in \Omega$ and $\eta\in \mathcal{C}_0^{\infty}(\Omega)$, $\eta(x_0)=1$, we obtain
\begin{equation}\label{minusinfty}
\alpha_n :=\|\nabla \tilde w_n\|^2_{L^2(\Omega)}\to +\infty,
\qquad
\frac{\int_{\Omega} |\tilde w_n|^{p+1} \,dx}{\alpha_n^{N(p-1)/4}}\to C_{N,p},
\qquad
\mathcal{E}_{\mu}(\tilde w_n)\to -\infty
\end{equation}
for $n\to +\infty$. Since $u_1$ is a local minimum,
the functional $\mathcal{E}_{\mu}$ has a mountain pass structure on ${\mathcal{M}}$;
by recalling that $\mathcal{E}_{\mu}$ satisfies the P.S. condition the
proposition follows.
\end{proof}
\begin{remark}\label{rem:further_crit_lev}
One can generalize Proposition \ref{mpcritlev} by constructing critical points via a saddle-point theorem in the following way: let us pick $k$ points $x_1, x_2,...,x_k$ in $\Omega$ and consider the corresponding function $\tilde w_i$; we may assume that
$\mathrm{supp}\,\tilde w_i\cap \mathrm{supp}\,\tilde w_j=\emptyset$ for $i\neq j$, so that these functions are orthogonal. Let us now define the subspace
$V_k=\mathrm{span}\{\varphi_1,\dots,\varphi_k;\tilde w_1,...,\tilde w_k\}$; note that dim $V_k=2k$. Let $R$ be an operator (in $L^2(\Omega)$) such that $R=I$ on $V_k^{\perp}$, $Ru_i=\tilde w_i$, $i=1,2,..,k$. Possibly after permutations, we can choose $R$ such that $R\big |_{V_k}\in SO(2k)$ (actually, there are infinitely many different choices of $R$). Now, since $SO(2k)$ is (arcwise) connected, there is a continuous path
$\tilde{\gamma}:\,[0,1]\rightarrow SO(2k)$ such that $\gamma(0)=I$, $\gamma(1)=R\big |_{V_k}$.
Then, we can define the following map
\[
\gamma:\, [0,1] \times S^{k-1}\rightarrow {\mathcal{M}},
\quad\quad \gamma(s;t_1,....,t_k)=\sum_{i=1}^{k}t_i\tilde{\gamma}(s)u_i,\quad
\]
where $\sum_{i=1}^{k}t_i^2=1$. It is clear that $\gamma$ is continuous; moreover,
\[
\gamma(0;t_1,....,t_k)\in \mathrm{span }\{\varphi_1,\dots,\varphi_k\}\cap {\mathcal{M}}
\quad \mathrm{and}\quad
\gamma(1;t_1,....,t_k)\in \mathrm{span }\{\tilde w_1,\dots,\tilde w_k\}\cap {\mathcal{M}}.
\]
Then, by denoting with $\Gamma_k$ the set of the above paths, if $\mu$ is sufficiently
small we obtain the critical levels
\[
\bar c_k : =\inf_{\gamma\in \Gamma_k}\sup_{[0,1]\times S^{k-1}}\mathcal{E}_{\mu}(\gamma(s;t_1,....,t_k)).
\]
\end{remark}
\section{Results in symmetric domains}\label{sec:symm}
This section is devoted to the proof of Theorem \ref{pro:symm}, therefore we assume $1+4/N \leq p < 2^*-1$.
We perform the proof in the case of $\Omega=B$, but it will be clear that the main assumption on $\Omega$ is the following:
\begin{itemize}
\item[\textbf{(T)}] there is a tiling of $\Omega$, made by $h$ copies of a subdomain $D$, in such a way that from
any solution $U_D$ of \eqref{eq:main_prob_U} on $D$ one can construct, using reflections, a solution $U_\Omega$ of \eqref{eq:main_prob_U} on $\Omega$.
\end{itemize}
Then $U_\Omega$ has $h$ times the mass of $U_D$, and recalling Theorem \ref{thm:intro_GS} we deduce that \eqref{eq:main_prob_U} on $\Omega$ is solvable for any
$\rho< h \cdot D_{N,p} \lambda_1(D)^{\frac{2}{p-1}-\frac{N}{2}}$. At this point, for a sequence $(D_k,h_k)_k$ of tilings satisfying \textbf{(T)}, we
obtain the solvability of \eqref{eq:main_prob_U} on $\Omega$ whenever
\[
\rho< h_k \cdot D_{N,p} \lambda_1(D_k)^{\frac{2}{p-1}-\frac{N}{2}},
\]
and if we can show that
\begin{equation}\label{eq:finaltarget}
\frac{ h_k }{ \lambda_1(D_k)^{\frac{N}{2}-\frac{2}{p-1}}} \to +\infty\qquad\text{as }k\to+\infty,
\end{equation}
we deduce the solvability of \eqref{eq:main_prob_U} on $\Omega$ for every mass. Having this scheme in mind, it is easy to prove analogous results
on rectangles and also in other kind of domains.
Then let $B\subset{\mathbb{R}}^N$ be the ball (w.l.o.g. of radius one), and let
\[
D_k:=\left\{(r\cos\theta, r\sin\theta,x_3,\dots,x_N)\in B: - \frac{\pi}{k} < \theta < \frac{\pi}{k}\right\}
\]
Then $D_k$ satisfies \textbf{(T)}, with $h_k=k$. In order to estimate $\lambda_1(D_k)$ we observe that, by elementary trigonometry,
\[
B'_k = B_{\frac{\sin(\pi/k)}{\sin(\pi/k)+1}}\left(\frac{1}{\sin(\pi/k)+1},0,0,\dots,0\right) \subset D_k,
\]
and therefore
\[
\lambda_1(D_k) \le \lambda_1(B'_k) \le C k^2,
\]
for some dimensional constant $C=C(N)$ and $k$ large. Then
\[
\frac{ h_k }{ \lambda_1(D_k)^{\frac{N}{2}-\frac{2}{p-1}}}\ge C \frac{k}{k^{{N}-\frac{4}{p-1}}} = C k^{1-{N}+\frac{4}{p-1}} = C k^{\frac{N-1}{p-1}\left[1+\frac{4}{N-1} - p\right]},
\]
and finally \eqref{eq:finaltarget} holds true whenever $p< 1+\frac{4}{N-1}$, thus completing the proof of Theorem \ref{pro:symm}.
\small
\subsection*{Acknowledgments}
We would like to thank Jacopo Bellazzini, who pointed out that the results in
\cite{MR3318740}, in the supercritical case, can be read in terms of a local minimization. We
would also like to thank Benedetta Noris, who read a preliminary version of this manuscript.
This work is partially supported by the PRIN-2012-74FYK7 Grant:
``Variational and perturbative aspects of nonlinear differential problems'',
by the ERC Advanced Grant 2013 n. 339958:
``Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT'',
and by the INDAM-GNAMPA group.
| {'timestamp': '2016-07-18T02:07:27', 'yymm': '1607', 'arxiv_id': '1607.04520', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04520'} | arxiv |
"\\section{Introduction}\n\\label{sec:intro}\nDespite the immense popularity and availability of onl(...TRUNCATED) | "{'timestamp': '2016-07-20T02:04:30', 'yymm': '1607', 'arxiv_id': '1607.04648', 'language': 'en', 'u(...TRUNCATED) | arxiv |
"\\section{Introduction}\n\nThe wave-particle duality is an alternative statement of the complementa(...TRUNCATED) | "{'timestamp': '2017-01-26T02:07:48', 'yymm': '1607', 'arxiv_id': '1607.04617', 'language': 'en', 'u(...TRUNCATED) | arxiv |
"\\section{#1}\\setcounter{equation}{0}}\n\\newcommand{\\subsect}[1]{\\subsection{#1}}\n\\renewcomma(...TRUNCATED) | "{'timestamp': '2016-07-19T02:06:15', 'yymm': '1607', 'arxiv_id': '1607.04824', 'language': 'en', 'u(...TRUNCATED) | arxiv |
"\\section{Introduction}\n\\label{sec:introduction} \nIn recent years there has been a resurgen(...TRUNCATED) | "{'timestamp': '2016-12-01T02:04:39', 'yymm': '1607', 'arxiv_id': '1607.04724', 'language': 'en', 'u(...TRUNCATED) | arxiv |
"\\section{Introduction} \\label{sec:intro}\n\nThe production of hadrons and jets at a future Electr(...TRUNCATED) | "{'timestamp': '2016-10-28T02:00:49', 'yymm': '1607', 'arxiv_id': '1607.04921', 'language': 'en', 'u(...TRUNCATED) | arxiv |
"\\section{Introduction}\n\\label{sec:intro}\n\n\nCompact stars have a large number of pulsation mod(...TRUNCATED) | "{'timestamp': '2016-07-19T02:02:47', 'yymm': '1607', 'arxiv_id': '1607.04707', 'language': 'en', 'u(...TRUNCATED) | arxiv |
End of preview.
Getting Started
The dataset consists of 2084 jsonl files. You can download the dataset using HuggingFace:
from datasets import load_dataset
ds = load_dataset("togethercomputer/RedPajama-Data-1T")
Or you can directly download the files using the following command:
wget 'https://data.together.xyz/redpajama-data-1T/v1.0.0/urls.txt'
while read line; do
dload_loc=${line#https://data.together.xyz/redpajama-data-1T/v1.0.0/}
mkdir -p $(dirname $dload_loc)
wget "$line" -O "$dload_loc"
done < urls.txt
After downloading the files, you can load the dataset from disk by setting the RED_PAJAMA_DATA_DIR environment variable to the directory containing the files:
import os
from datasets import load_dataset
os.environ["RED_PAJAMA_DATA_DIR"] = "/path/to/download"
ds = load_dataset("togethercomputer/RedPajama-Data-1T")
A smaller 1B-token sample of the dataset can be found here.
A full set of scripts to recreate the dataset from scratch can be found here.
Dataset Summary
RedPajama is a clean-room, fully open-source implementation of the LLaMa dataset.
| Dataset | Token Count |
|---|---|
| Commoncrawl | 878 Billion |
| C4 | 175 Billion |
| GitHub | 59 Billion |
| ArXiv | 28 Billion |
| Wikipedia | 24 Billion |
| StackExchange | 20 Billion |
| Total | 1.2 Trillion |
Languages
Primarily English, though the Wikipedia slice contains multiple languages.
Dataset Structure
The dataset structure is as follows:
{
"text": ...,
"meta": {"url": "...", "timestamp": "...", "source": "...", "language": "...", ...},
"red_pajama_subset": "common_crawl" | "c4" | "github" | "arxiv" | "wikipedia" | "stackexchange"
}
Dataset Creation
This dataset was created to follow the LLaMa paper as closely as possible to try to reproduce its recipe.
Source Data
Commoncrawl
We download five dumps from Commoncrawl, and run the dumps through the official cc_net pipeline.
We then deduplicate on the paragraph level, and filter out low quality text using a linear classifier trained to
classify paragraphs as Wikipedia references or random Commoncrawl samples.
C4
C4 is downloaded from Huggingface. The only preprocessing step is to bring the data into our own format.
GitHub
The raw GitHub data is downloaded from Google BigQuery. We deduplicate on the file level and filter out low quality files and only keep projects that are distributed under the MIT, BSD, or Apache license.
Wikipedia
We use the Wikipedia dataset available on Huggingface, which is based on the Wikipedia dump from 2023-03-20 and contains text in 20 different languages. The dataset comes in preprocessed format, so that hyperlinks, comments and other formatting boilerplate has been removed.
Gutenberg and Books3
Defunct: The 'book' config is defunct and no longer accessible due to reported copyright infringement for the Book3 dataset contained in this config.
ArXiv
ArXiv data is downloaded from Amazon S3 in the arxiv requester pays bucket. We only keep latex source files and
remove preambles, comments, macros and bibliographies.
Stackexchange
The Stack Exchange split of the dataset is download from the Internet Archive. Here we only keep the posts from the 28 largest sites, remove html tags, group the posts into question-answer pairs, and order answers by their score.
SHA256 Checksums
SHA256 checksums for the dataset files for each data source are available here:
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/arxiv_SHA256SUMS.txt
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/c4_SHA256SUMS.txt
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/common_crawl_SHA256SUMS.txt
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/github_SHA256SUMS.txt
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/stackexchange_SHA256SUMS.txt
https://data.together.xyz/redpajama-data-1T/v1.0.0/sha256/wikipedia_SHA256SUMS.txt
To cite RedPajama, please use:
@software{together2023redpajama,
author = {Together Computer},
title = {RedPajama: An Open Source Recipe to Reproduce LLaMA training dataset},
month = April,
year = 2023,
url = {https://github.com/togethercomputer/RedPajama-Data}
}
License
Please refer to the licenses of the data subsets you use.
- Common Crawl Foundation Terms of Use
- C4 license
- GitHub was limited to MIT, BSD, or Apache licenses only
- ArXiv Terms of Use
- Wikipedia License
- StackExchange license on the Internet Archive
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