\documentclass[reqno]{amsart}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 81, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/81\hfil Concentration and dynamic system]
{Concentration and dynamic system of solutions for semilinear
elliptic equations}

\author[Tsung-fang Wu\hfil EJDE--2003/81\hfilneg]
{Tsung-fang Wu}

\address{Tsung-fang Wu \newline
Center for General Education\\
Southern Taiwan University of Technology, Taiwan}
\email{tfwu@mail.stut.edu.tw}



\date{}
\thanks{Submitted June 23, 2003. Published August 7, 2003.}
\thanks{Partially supported by the National Science Council of China}
\subjclass[2000]{35J20, 35J25, 35J60}
\keywords{Palais-Smale, concentration, dynamic system, multiple solutions}

\begin{abstract}
  In this article, we  use the concentration of solutions of
  the semilinear elliptic equations in axially symmetric bounded
  domains to prove that the equation has three positive solutions.
  One solution is $y$-symmetric and the other are non-axially
  symmetric. We also study the dynamic system of these
  solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Consider the semilinear elliptic equation
\begin{equation}
\begin{gathered}
-\Delta u+u=|u|^{p-2}u \quad\text{in }\Omega , \\
u\in H_{0}^{1}(\Omega ),
\end{gathered} \label{E1}
\end{equation}
where $N\geq 2$, $2^{*}=\frac{2N}{N-2}$ for $N\geq 3$ and $2^{*}=\infty $
for $N=2$, $2<p<2^{*}$, $\Omega $ is a domain in $\mathbb{R}^{N}$, and 
$H_{0}^{1}(\Omega) $ is the Sobolev space in $\Omega $ with dual
space $H^{-1}(\Omega )$. Associated with equation (\ref{E1}), we consider
the energy functionals $a$, $b$, and $J$ defined for each
$u\in H_{0}^{1}(\Omega )$ as follows:
\begin{gather*}
a(u) =\int_{\Omega }(|\nabla u|^{2}+u^{2}), \quad
b(u) =\int_{\Omega }|u|^{p}, \\
J(u) =\frac{1}{2}a(u)-\frac{1}{p}b(u).
\end{gather*}
By Rabinowitz \cite[Proposition B. 10]{R}, $a$, $b$, and $J$ are of class 
$C^{1,1}$. It is well-known that the solutions of equation (\ref{E1}) are the
critical points of the energy functional $J$. Let
$z=(x,y)\in \mathbb{R}^{N-1}\times \mathbb{R}$. Denote the $N$-ball $B^{N}(z_{0};s)$
in $\mathbb{R}^{N}$, the infinite strip $\mathbf{A}^{r}$, the upper half strip
$\mathbf{A}_{0}^{r}$, and the finite strip $\mathbf{A}_{s,t}^{r}$ as follows:
\begin{gather*}
B^{N}(z_{0};s)=\{z\in \mathbb{R}^{N}:|z-z_{0}|<s\}, \\
\mathbf{A}^{r}=\{(x,y)\in \mathbb{R}^{N}:|x|<r\}, \\
\mathbf{A}_{0}^{r}=\{(x,y)\in \mathbf{A}^{r}:0<y\}, \\
\mathbf{A}_{s,t}^{r}=\{(x,y)\in \mathbf{A}^{r}:s<y<t\}.
\end{gather*}
We should point out here that the precise definition of the finite
strip $\mathbf{A}_{s,t}^{r}$ is the domain which is symmetric in
$y-$axis and has been smoothed out at the corners of $\{(x,y)\in
\mathbf{A}^{r}:s<y<t\}$. By the Rellich compactness theorem, there
is a positive solution of equation (\ref{E1}) in the finite strip
$\mathbf{A}_{-t,t}^{r}$ for each $t>0$. Moreover,
$\mathbf{A}_{-t,t}^{r}$ is convex in $x$ and in $y$. Thus, by
Gidas-Ni-Nirenberg \cite{GNN}, every positive solution of equation
(\ref{E1}) in $\mathbf{A}_{-t,t}^{r}$ for each $t>0$ is radially
symmetric in $x$ and axially symmetric in $y$. Actually, Dancer
\cite{Da} proved that the positive solution of equation (\ref{E1})
in $\mathbf{A}_{-t,t}^{r}$ for each $t>0$ in $\mathbb{R}^{2}$ is
unique. However, the axially symmetry and uniqueness of positive
solution generally fails if $\Omega $ is not convex in the
$y$-direction. First, we consider a perturbation of the finite
strip $\mathbf{A}_{-t,t}^{r}$, that is dumbbell type domain
\[
D=B^{N}((0;-t) ,r_{0}) \cup \mathbf{A}_{-t,t}^{r}\cup B^{N}((0;t) ,r_{0})
\quad \quad\text{for } B^{N-1}(0;r) \subset B^{N-1}(0;r_{0}) .
\]
Then the dumbbell domain $D$ is symmetric in $y-$axis, but not
convex in $y$-direction. Moreover, the Dancer \cite{Da} and Byeon
\cite{By1}, \cite{By2} proved that the equation $(\ref{E1}) $ in $D$ has at least three positive solutions,
for $B^{N-1}(0;r) $ is sufficiently close to a point
$x_{0}$ in $\mathbb{R}^{N-1}$. And Chen-Ni-Zhou \cite{CNZ} use
computational showed that the equation $(\ref{E1}) $
in some dumbbell-type domains has multiple positive solutions and
describe the concentration of these solutions.

The main purpose of this paper is using the Palais-Smale theory to
present another perturbation. Let $\omega $ be a $y$-symmetric
bounded set such that $\mathbf{A}^{r}\backslash \overline{\omega
}\subsetneqq \mathbf{A}^{r}$ is a domain in $\mathbb{R}^{N}$ for some
$t>0$, consider the finite strip with holes
\[
\Theta _{t}=\mathbf{A}_{-t,t}^{r}\backslash \overline{\omega }.
\]
Then there exists a $t'>0$ such that $\Theta _{t}$ is
also symmetric in $y-$axis, but not convex in $y-$direction for
each $t>t'$. We prove that there exists a $t_{0}>0$ such
that for $t\geq t_{0}$, the equation $(\ref{E1}) $ in
$\Theta _{t}$ has three positive solutions which one is
$y$-symmetric and the other are non-axially symmetric. Moreover,
we describe the concentration and dynamic system of these
solutions. Although, Wang-Wu \cite{WW} used the symmetry of
positive
solutions showed the same multiple results in a finite strip with hole
$\mathbf{A}_{-t,t}^{r}\backslash B^{N}(0;r')$ for $t$ sufficiently
large. However, they have not describe the concentration and
dynamic system of solutions.

This article is organized as follow. In section 2, we describe various
preliminaries. In section 3, we describe various compactness results. In
section 4, we describe some properties of the large domains in $\mathbf{A}^{r}$.
In section 5 and section 6, we present the concentration and dynamic
system of the solutions.

\section{Preliminary}

In this article, we focus on the problems on two Hilbert spaces: the whole
Sobolev space $H_{0}^{1}(\Omega )$ and its closed linear subspace
$H_{s}(\Omega )$ defined as follows: Let
$z=(x,y)\in \mathbb{R}^{N-1}\times \mathbb{R}$ and $\Omega $ be a domain in $\mathbb{R}^{N}$.

\begin{definition} \label{p1} \rm
\begin{itemize}
\item[(i)] $\Omega $ is $y$-symmetric provided $\,z=(x,y)\in \Omega $
if and only if $(x,-y)\in \Omega $;
\item[(ii)]  Let $\Omega $ be a $y$-symmetric domain in $\mathbb{R}^{N}$.
A function $u:\Omega \to \mathbb{R}$ is $y$-symmetric (axially symmetric) if
$u(x,y)=u(x,-y)$ for $(x,y)\in \Omega$.
\end{itemize}
\end{definition}

 In  this article, we let $\Omega $ be a $y$-symmetric domain
in $\mathbb{R}^{N}$ and $H_{s}(\Omega )$ the $H^{1}$-closure of the space
$\{u\in C_{0}^{\infty }(\Omega ) : u \quad\text{is $y$-symmetric}\}$ and let
$X(\Omega) $ be either the whole space $H_{0}^{1}(\Omega) $ or the $y$-symmetric
Sobolev space $H_{s}(\Omega)$.
Then $H_{s}(\Omega )$ is a closed linear subspace of $H_{0}^{1}(\Omega )$.
Let $H_{s}^{-1}(\Omega) $ be the dual space of $H_{s}(\Omega )$.

 We define the Palais-Smale (simply by (PS)) sequences, (PS)-values and
 (PS)-conditions in $X(\Omega) $ for $J$ as follows.


\begin{definition} \label{p2} \rm
\begin{itemize}
\item[(i)]
 For $\beta \in \mathbb{R}\mathbf{,}$ a sequence $\{ u_{n}\} $
is a (PS)$_{\beta }$-sequence in $X(\Omega) $ for $J$ if $%
J(u_{n})=\beta +o(1)\;$and$\;J'(u_{n})=o(1)\;$strongly in $%
X^{-1}(\Omega) $ as $n\to \infty $
\item[(ii)] $\beta \in \mathbb{R}$ is a (PS)-value in $X(\Omega) $
for $J$ if there is a (PS)$_{\beta }$-sequence in $X(\Omega) $
for $J$
\item[(iii)] $J$ satisfies the (PS)$_{\beta }-$condition in $X(\Omega) $
if every (PS)$_{\beta }$-sequence in $X(\Omega) $ for $J$ contains a convergent
subsequence.
\end{itemize}
\end{definition}

 Now, we consider the Nehari minimization problem
\[
\alpha _{X}(\Omega )=\inf_{u\in \mathbf{M}(\Omega )}J(u),
\]
where $\mathbf{M}(\Omega )=\left\{ u\in X(\Omega)
\backslash \{0\}:a(u)=b(u)\right\} $. Note that
$\mathbf{M}(\Omega )$ contains every nonzero solution of equation
(\ref{E1}) in $\Omega $, $\alpha _{X}(\Omega )>0$, and if 
$u_{0}\in \mathbf{M}(\Omega )$ achieves $\alpha _{X}(\Omega )$, then $u_{0}$
is a positive (or negative) solution of equation (\ref{E1}) in
$\Omega $ (see Wang-Wu \cite{WW} or Willem \cite{Wi}).
 We have the following useful lemma, whose proof can be found in
Wang-Wu \cite[Lemma 7]{WW}.

\begin{lemma} \label{p7}
Let $\{u_{n}\}$ be in $X(\Omega) $. Then $\{u_{n}\}$
is a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega) $ for
$J$ if and only if $J(u_{n}) =\alpha _{X}(\Omega )+\mathrm{o}%
(1) $ and $a(u_{n}) =b(u_{n}) +\mathrm{o}%
(1) $. In particular, every minimizing sequence $\{u_{n}\}$ in 
$\mathbf{M}(\Omega )$ for $\alpha _{X}(\Omega )$ is a \newline
(PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega )$ for $J$.
\end{lemma}

We denote $\alpha _{X}(\Omega )$ by $\alpha (\Omega )$ for
$X(\Omega) =H_{0}^{1}(\Omega) $. We denote $\alpha
_{X}(\Omega )$ by $\alpha _{s}(\Omega )$ for $X(\Omega)
=H_{s}(\Omega)$.
We denote $\mathbf{M}(\Omega )$ by $\mathbf{M}_{0}(\Omega )$ for
$X(\Omega) =H_{0}^{1}(\Omega) $.  We denote $\mathbf{M}(\Omega )$ by
$\mathbf{M}_{s}(\Omega )$ for $X(\Omega) =H_{s}(\Omega) $.

\begin{remark} \rm
By the Principle of symmetric criticality (see Palais \cite{P}), we have a
(PS)$_{\beta }$-sequence in $X(\Omega) $ for $J$ is a (PS)$%
_{\beta }$-sequence in $H_{0}^{1}(\Omega) $ for $J$.
\end{remark}

\section{Palais-Smale Conditions}

In this section, we present several (PS)$_{\alpha _{X}(\Omega) }-$conditions in $X(\Omega )$ for $J$ which are
used to prove our main results in section 4 and section 5. Since
for each (PS)$_{\alpha _{X}(\Omega) }$-sequence
$\{ u_{n}\} $ in $X(\Omega
) $ for $J$, there exists a subsequence $\{ u_{n}\} $ and $u$ in $%
X(\Omega )$ such that $u_{n}\rightharpoonup u$ weakly in $X(\Omega )$. Then 
$u$ is a solution of equation (\ref{E1}) in $\Omega $. Moreover, we
have the following result, whose proof can be found in
Bahri-Lions \cite{BL} and in Wang-Wu \cite{WW}.

\begin{lemma} \label{c4}
For each (PS)$_{\alpha _{X}(\Omega) }$-sequence 
$\{u_{n}\}$ in $X(\Omega) $ for $J$, there exists a subsequence 
$\{u_{n}\}$and a nonzero $u$ in $X(\Omega )$ such that
$u_{n}\rightharpoonup u $ weakly in $X(\Omega) $ if
and only if the (PS)$_{\alpha _{X}(\Omega)
}-$condition holds in $X(\Omega )$ for $J$.
\end{lemma}

 Let $\Omega $ be any unbounded domain and $\xi \in C^{\infty}([0,\infty ))$
such that $0\leq \xi \leq 1$ and
\[
\xi (t)=\begin{cases}
0, & \text{for }t\in [0,1] \\
1, & \text{for }t\in [2,\infty ).
\end{cases}
\]
Let
\begin{equation}
\xi _{n}(z)=\xi (\frac{2|z|}{n}).  \label{11}
\end{equation}
Then we have the following results.

\begin{proposition} \label{c1}
The equation $(\ref{E1}) $ in $\Omega $ does not admit
any solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega )$
if and only if for each (PS)$_{\alpha _{X}(\Omega )}$-sequence
$\{u_n\} $ in $X(\Omega) $ for $J$, there exists a
subsequence $\{u_n\} $ such that $\{ \xi_{n}u_{n}\} $ is also a
(PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega) $ for $J$.
\end{proposition}

\begin{proof}
Let $\{u_n\} $ be a (PS)$_{\alpha _{X}(\Omega )}$-sequence in
$X(\Omega) $ for $J$. Then there exist a subsequence $\{u_n\} $ and
$u_{0}\in X(\Omega) $ such that $u_{n}\rightharpoonup u_{0}$
weakly in $X(\Omega) $.
Since the equation $(\ref{E1}) $ in $\Omega $ does not
admit any solution $u_{0}$ such that $J(u_{0}) =\alpha
_{X}(\Omega )$, by Lemma \ref {c4}, we have $u_{0}=0$. Let
$v_{n}=\xi _{n}u_{n}$. First, we need to show
\begin{equation}
a(u_{n}-v_{n}) =o(1) .  \label{11-2}
\end{equation}
Note that
\[
a(u_{n}-v_{n}) =a(u_{n}) +a(v_{n})
-2\left\langle u_{n},v_{n}\right\rangle _{H^{1}}.
\]
Thus, it suffices to show that $\left\langle u_{n},v_{n}\right\rangle
_{H^{1}}=a(u_{n}) +o(1) =a(v_{n})+o(1) $. Since
\begin{align*}
\left\langle u_{n},v_{n}\right\rangle _{H^{1}}
&=\int_{\Omega }\nabla u_{n}\nabla v_{n}+u_{n}v_{n} \\
&=\int_{\Omega }\xi _{n}\left[ | \nabla u_{n}|
^{2}+u_{n}^{2}\right] +\int_{\Omega }u_{n}\nabla u_{n}\nabla \xi _{n}.
\end{align*}
Note that $|\nabla \xi _{n}| \leq \frac{c}{n}$ and $\{u_n\} $ is a
(PS)$_{\alpha _{X}(\Omega ) }$-sequence in $X(\Omega) $ for $J$,
so
\begin{equation}
\int_{\Omega }\xi _{n}^{q}u_{n}\nabla u_{n}\nabla \xi _{n}=o(1)
\quad \text{for }q>0.  \label{12}
\end{equation}
Hence,
\begin{equation}
\left\langle u_{n},v_{n}\right\rangle _{H^{1}}=\int_{\Omega }\xi _{n}\left[
| \nabla u_{n}| ^{2}+u_{n}^{2}\right] +o(1) .
\label{13}
\end{equation}
Similarly, we have
\begin{equation}
a(v_{n}) =\int_{\Omega }\xi _{n}^{2}\left[ | \nabla
u_{n}| ^{2}+u_{n}^{2}\right] +o(1) .  \label{14}
\end{equation}
For $r\geq 1$. Since $\{\xi _{n}^{r}u_{n}\}$ is bounded in $X(\Omega )$, we
have
\begin{align*}
o(1) &=\left\langle J'(u_{n}),\xi _{n}^{r}u_{n}\right\rangle \\
&=\int_{\Omega }(\xi _{n}^{r}|\nabla u_{n}|^{2}+r\xi
_{n}^{r-1}u_{n}\nabla \xi _{n}\nabla u_{n}+\xi
_{n}^{r}u_{n}^{2})-\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}.
\end{align*}
By $(\ref{12})$, we conclude that
\begin{equation}
\int_{\Omega }\xi _{n}^{r}(|\nabla
u_{n}|^{2}+u_{n}^{2})=\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}+o(1).  \label{15}
\end{equation}
Since $u_{n}\rightharpoonup 0$ weakly in $H_{0}^{1}(\Omega )$, there exists
a subsequence $\{u_n\} $ such that $u_{n}\to 0$
strongly in $L_{loc}^{p}(\Omega) $, or there exists a
subsequence $\{u_n\} $ such that
\[
\int_{Q(n) }|u_n| ^{p}=o(1),
\]
where $Q(n) =\Omega \cap B^{N}(0;n) $. Clearly,
\begin{equation}
\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}=\int_{\Omega
}|u_{n}|^{p}+o(1).  \label{15-1}
\end{equation}
By $(\ref{13}) ,(\ref{14}), (\ref{15})$
and $(\ref{15-1})$, we have
\[
\left\langle u_{n},v_{n}\right\rangle _{H^{1}}=a(u_{n}) +o(1) =a(v_{n}) +o(1) .
\]
Moreover, by the compact imbedding theorem, we obtain
\begin{equation}
b(v_{n}) =b(u_{n}) +o(1) .  \label{15-2}
\end{equation}
Since $a(u_{n}) =b(u_{n}) +o(1) $. Thus, from $(\ref{11-2})$
and $(\ref{15-2})$, we obtain
\begin{gather*}
a(v_{n}) =b(v_{n}) +(1)\,,\
J(v_{n}) =\alpha _{X}(\Omega )+o(1) \,.
\end{gather*}
By Lemma \ref{p7}, we can conclude that $\left\{ \xi _{n}u_{n}\right\} $ is
a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega) $ for $%
J. $ Conversely, assume that the equation $(\ref{E1}) $ in $%
\Omega $ admits a solution $u_{0}$ such that $J(u_{0}) =\alpha
_{X}(\Omega )$. We may assume that $u_{0}$ is a positive solution. Let $%
u_{n}=u_{0}$ for each $n\in \mathbb{N}$, then $\{u_n\} $ is a (PS)%
$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega) $ for $J$. By
hypothesis, we have $\left\{ \xi _{n}u_{0}\right\} $ is also a (PS)$_{\alpha
_{X}(\Omega )}$-sequence in $X(\Omega) $ for $J$. We obtain
\[
\int_{\Omega }| \xi _{n}u_{0}| ^{p}=\frac{2p}{p-2}\alpha
_{X}(\Omega )+o(1) .
\]
Thus, there exist $n_{0}$ and $d>0$ such that
\begin{equation}
\int_{\Omega }| \xi _{n}u_{0}| ^{p}>d\quad\text{for each }n\geq n_{0}.
\label{15-3}
\end{equation}
However, $u_{0}\in L^{p}(\Omega) $. Hence
\[
\int_{\Omega }| \xi _{n}u_{0}| ^{p}\leq \int_{\left[ B^{N}(0;\frac{n}{2}) \right] ^{c}}| u_{0}| ^{p}=o(1)
\quad\text{as }n\to \infty ,
\]
this contradicts to $(\ref{15-3}) $.
\end{proof}

\begin{proposition} \label{c2}
$J$ does not satisfy the (PS)$_{\alpha _{X}(\Omega
)}-$condition
in $X(\Omega) $ for $J$ if and only if there exists a (PS)$%
_{\alpha _{X}(\Omega )}$-sequence $\{u_{n}\}$ in $X(\Omega) $ for $J$ such that 
$\left\{ \xi _{n}u_{n}\right\} $ is
also a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega) $ for $J$.
\end{proposition}

The proof of this proposition is similar to the proof of Proposition \ref{c1}
and therefore, it is omitted.

 Let $\Omega _{1}\subsetneqq \Omega _{2}$, clearly $\alpha
_{X}(\Omega _{1}) \geq \alpha _{X}(\Omega _{2}) $.
Then we have the following useful results.

\begin{lemma}\label{c5}
Let $\Omega _{1}\subsetneqq \Omega _{2}$ and $J:X(\Omega
_{2})\to \mathbb{R}$ be the energy functional. Suppose that
$\alpha
_{X}(\Omega _{1}) =\alpha _{X}(\Omega _{2}) $. Then
\begin{itemize}
\item[(i)] The equation $(\ref{E1}) $ in $\Omega _{1}$ does
not admit any solution $u_{0}$ such that $J(u_{0})
=\alpha _{X}(\Omega _{1})$
\item[(ii)] $J$ does not satisfy
the (PS)$_{\alpha _{X}(\Omega _{2})}-$condition.
\end{itemize}
\end{lemma}

The proof of this lemma can be found in Wang-Wu \cite[Lemma 13]{WW}.
By the Rellich compact theorem, $J$ satisfies the (PS)$_{\alpha
_{X}(\Omega )}-$condition in $X(\Omega )$ if $\Omega $ is a
bounded domain.

\begin{lemma} \label{c8}
Let $\Omega $ be a bounded domain in $\mathbb{R}^{N}$. Then the (PS)$%
_{\alpha _{X}(\Omega )}-$condition holds in $X(\Omega )$ for $J$.
Furthermore, the equation $(\ref{E1}) $ in $\Omega $ has a
positive solution $u_{0}$ such that $J(u_{0}) =\alpha
_{X}(\Omega )$.
\end{lemma}

\section{Large Domains in $\mathbf{A}^r$}

\begin{definition} \label{e1} \rm
A domain $\Omega $ in $\mathbf{A}^{r}$ is large if for any $m>0$,
there exist $s<t$ such that $t-s=m$ and
$\mathbf{A}_{s,t}^{r}\subset \Omega $.
\end{definition}

\begin{lemma} \label{e4}
If $\Omega $ is a large domain in $\mathbf{A}^{r}$, then $\alpha
(\Omega )=\alpha (\mathbf{A}^{r})$. Furthermore, if $\Omega $ is a
proper large domain in $\mathbf{A}^{r}$, then the equation
$(\ref{E1}) $ in $\Omega $ does not admit any solution $u_{0}$
such that $J(u_{0}) =\alpha (\Omega )$.
\end{lemma}

The proof of this lemma follows by Lien-Tzeng-Wang \cite[Lemma 2.5]{LTW} and
Lemma \ref{c5}.


 We need the following symmetric result to assert our main result.

\begin{lemma}\label{e6}
Suppose that $\Omega $ is a $y$-symmetric large domain in $\mathbf{%
A}^{r}$. Then $\alpha _{s}(\Omega) \leq 2\alpha (\mathbf{A%
}^{r}) $.
\end{lemma}

\begin{proof}
Since $\Omega $ is a $y$-symmetric large domain in
$\mathbf{A}^{r}$. Thus, there exist $t_{0}>0$, $\Omega _{1}$
 and $\Omega _{2}$ are large domains in
$\mathbf{A}^{r}$ such that $\Omega \backslash \overline{\mathbf{A}%
_{-t_{0},t_{0}}^{r}}=\Omega _{1}\cup \Omega _{2}$. Let $\left\{
u_{n}^{1}\right\} $ be a (PS)$_{\alpha (\Omega _{1})
}$-sequence in $H_{0}^{1}(\Omega _{1}) $ for $J$ and
let $u_{n}^{2}(x,y) =u_{n}^{1}(x,-y) $.
Clearly, $\left\{ u_{n}^{2}\right\} $ is a (PS)$_{\alpha (\Omega _{2}) }$-sequence
in $H_{0}^{1}(\Omega _{2}) $ for $J$. Take $v_{n}=u_{n}^{1}+u_{n}^{2}$,
then $v_{n}\in H_{s}(\Omega)$, $a(v_{n}) =b(v_{n}) +o(1)$ and
\[
J(v_{n}) =\alpha (\Omega _{1}) +\alpha
(\Omega _{2}) +o(1).
\]
Moreover, there exists $s_{n}>0$ such that $s_{n}v_{n}\in
\mathbf{M}_{s}(\Omega) $ and
\[
J(s_{s}v_{n}) =\alpha (\Omega _{1})
+\alpha (\Omega _{2}) +o(1).
\]
From Lemma \ref{e4} and the definition of Nehari minimization problem, we
can conclude $\alpha _{s}(\Omega) \leq 2\alpha (\mathbf{A}^{r}) $.
\end{proof}

 Then we have the following symmetric compactness.

\begin{proposition}\label{e7}
Suppose that $\Omega $ is a $y$-symmetric large domain in $\mathbf{A}^{r}$.
Then $J$ satisfies the (PS)$_{\alpha _{s}(\Omega )}-$condition in $%
H_{s}(\Omega) $ if and only if $\alpha _{s}(\Omega) <2\alpha (\mathbf{A}^{r}) $.
\end{proposition}

\begin{proof}
Suppose that $J$ satisfies the (PS)$_{\alpha _{s}(\Omega )}-$condition in $%
H_{s}(\Omega )$. By Lemma \ref{e6}, we have
$\alpha _{s}(\Omega) \leq 2\alpha (\mathbf{A}^{r}) $. Suppose that
$\alpha_{s}(\Omega) =2\alpha (\mathbf{A}^{r}) $. By the
definition of domain in $\mathbb{R}^{N}$, we may take a domain
$\tilde{\Omega}=\Omega \backslash \overline{B^{N}(0;\tilde{r}) }$ for some
$\tilde{r}>0$ such that $\tilde{\Omega}\subsetneqq \Omega $ and
$\tilde{\Omega}$ is a proper $y$-symmetric large domain in $\mathbf{A}^{r}$.
By Lemma \ref{c5}, we have
$2\alpha (\mathbf{A}^{r}) =\alpha _{s}(\Omega) <\alpha _{s}(\tilde{\Omega}) $.
This contradicts
to Lemma \ref{e6}. Conversely, suppose that $J$ does not satisfy the
(PS)$_{\alpha _{s}(\Omega )}-$condition. By Proposition \ref{c2}, there
exists a (PS)$_{\alpha _{s}(\Omega) }$-sequence
$\{u_n\} $ in $H_{s}(\Omega) $ for $J$
such that $\left\{ \xi_{n}u_{n}\right\} $ is also a
(PS)$_{\alpha _{s}(\Omega) }$-sequence in $H_{s}(\Omega) $ for $J$,
where $\xi_{n}$ is as in (\ref{11}). Let $v_{n}=\xi _{n}u_{n}$, we obtain
\begin{equation}
\begin{gathered}
J(v_{n})=\alpha _{s}(\Omega )+\mathrm{o}(1)\text{,} \\
J'(v_{n})=\mathrm{o}(1)\quad\text{in }H^{-1}(\Omega) .
\end{gathered}
\label{7}
\end{equation}
Since $\Omega $ is a $y$-symmetric large domain in
$\mathbf{A}^{r}$, there
exists a $n_{0}\in \mathbb{N}$ such that $v_{n}=0$ in $\overline{\Omega _{n_{0}}%
}$ for $n>2n_{0}$, and two disjoint subdomains $\Omega _{1}$ and
$\Omega _{2} $ such that
\begin{gather*}
(x,y)\in \Omega _{2}\quad\text{if and only if}\quad (x,-y)\in \Omega _{1}, \\
\Omega \ \backslash \ \overline{\Omega _{n_{0}}}=\Omega _{1}\cup
\Omega _{2},
\end{gather*}
where $\Omega _{n}=\left\{ z\in \Omega :-n<y<n\right\} $. Note that
$\Omega _{1}$ and $\Omega _{2}$ are also large domains in
$\mathbf{A}^{r}$.
Moreover, $v_{n}=v_{n}^{1}+v_{n}^{2}$ and for $i=1,2$,
\[
v_{n}^{i}(z)=\begin{cases}
v_{n}(z), & \text{for }z\in \Omega _{i} \\
0, & \text{for }z\notin \Omega _{i},
\end{cases}
\]
this implies $v_{n}^{i}\in H_{0}^{1}(\Omega _{i})$. By (\ref{7}),
we obtain
\[
J'(v_{n}^{i})=\mathrm{o}(1)\quad\text{strongly in }H^{-1}(\Omega _{i})%
\quad\text{for }i=1,2.
\]
We have $v_{n}^{1}(x,y) =v_{n}^{2}(x,-y)
,J(v_{n}^{1}) =J(v_{n}^{2}) $ and
\[
\alpha _{s}(\Omega) +o(1) =J(v_{n}) =J(v_{n}^{1}) +J(v_{n}^{2})
=2J(v_{n}^{i}) \quad\text{for }i=1,2,
\]
or
\[
J(v_{n}^{i}) =\frac{1}{2}\alpha _{s}(\Omega) +o(1) \quad\text{for }i=1,2.
\]
Therefore, $\frac{1}{2}\alpha _{s}(\Omega) $ is a positive (PS)-value in
$H_{0}^{1}(\Omega _{i})$ for $J$. By the definition of
Nehari minimization problem and Lemma \ref{e4}, we have
\[
\frac{1}{2}\alpha _{s}(\Omega) \geq \alpha (\Omega _{i}) =\alpha (\mathbf{A}^{r}) ,
\]
which is a contradiction.
\end{proof}

\begin{corollary} \label{e9}
Suppose that $\Omega $ is a $y$-symmetric large domain in
$\mathbf{A}^{r}$. Then $\alpha (\Omega) =\alpha _{s}(\Omega) $ if and only
if the equation $(\ref{E1}) $ in $\Omega $
has a $y$-symmetric solution $u_{0}$ such that $J(u_{0}) =\alpha(\Omega) $.
\end{corollary}

\begin{proof}
By Lemma \ref{e4}, we have
\[
\alpha (\mathbf{A}^{r}) =\alpha (\Omega) =\alpha
_{s}(\Omega) <2\alpha (\mathbf{A}^{r}) .
\]
By Proposition \ref{e7}, $J$ satisfies the (PS)$_{\alpha _{s}(\Omega )}-$%
condition in $H_{s}(\Omega )$. Thus, there exists a $y$-symmetric positive
solution $u_{0}$ such that
\[
J(u_{0}) =\alpha _{s}(\Omega) =\alpha (\Omega) .
\]
Conversely, use the definition of the Nehari minimization problem.
\end{proof}

\begin{proposition} \label{e10}
Suppose that $\Omega $ is a $y$-symmetric large domain in $\mathbf{A}^{r}$
such that $\alpha _{s}(\Omega) =2\alpha(\mathbf{A}^{r}) $. If
$\widetilde{\Omega }\subsetneqq \Omega $ is also $y$-symmetric large domain in
$\mathbf{A}^{r}$, then $\alpha _{s}(\widetilde{\Omega }) =2\alpha (\mathbf{A}^{r}) $ and the equation $(\ref{E1}) $
in $\widetilde{\Omega }$ does not admit any solution $u_{0}$ such
that $J(u_{0}) =\alpha _{s}(\widetilde{\Omega}) $.
\end{proposition}

The proof of this proposition follows from Lemma \ref{c5} and
Lemma \ref{e6}.

\begin{remark} \label{e11} \rm
From Lemma \ref{e6}, Proposition \ref{e7} and
Proposition \ref {e10}, the $y$-symmetric large domains in
$\mathbf{A}^{r}$ can be classify
into three kinds. If $\Omega $ is a $y$-symmetric large domain in $\mathbf{A%
}^{r}$, then it satisfies one of the following conditions:
\begin{enumerate}
\item %(\Omega 1)
$\alpha _{s}(\Omega) <2\alpha (\mathbf{A}^{r})$
\item % (\Omega 2)
$\alpha _{s}(\Omega) =2\alpha (\mathbf{A}^{r}) $ and the equation (\ref{E1})
in $\Omega $ has a solution $u_{0}$ such that $J(u_{0}) =\alpha _{s}(\Omega)$
\item  %(\Omega3)
$\alpha _{s}(\Omega) =2\alpha (\mathbf{A}^{r})$ and the equation
(\ref{E1}) in $\Omega $ does not admit any solution $u_{0}$ such that $J(u_{0})
=\alpha _{s}(\Omega) $.
\end{enumerate}
\end{remark}

\section{Concentration of Solutions}

For the rest of this article, let $\omega $ be a $y$-symmetric bounded
set such that $\mathbf{A}^{r}\backslash \overline{\omega }$ is a
$y$-symmetric proper large domain in $\mathbf{A}^{r}$. We need the
following notation:
\begin{gather*}
\mathbf{S} =\mathbf{A}^{r}\backslash \overline{\omega }; \\
\mathbf{S}_{k,l} =\left\{ (x,y) \in \mathbf{S} :k<y<l\right\} ; \\
\mathbf{S}_{-l}^{+} =\left\{ (x,y) \in \mathbf{S}:y\geq -l\right\} ; \\
\mathbf{S}_{l}^{-} =\left\{ (x,y) \in \mathbf{S}:y\leq l\right\} .
\end{gather*}
Note that $\mathbf{S,S}_{-l}^{+}$ and $\mathbf{S}_{l}^{-}$ are
proper large
domains in $\mathbf{A}^{r}$ for all $l\geq 0$. By Lemma \ref{e4}, we have $%
\alpha (\mathbf{S}) =\alpha (\mathbf{A}^{r}) $ and
the equation $(\ref{E1}) $ in $\mathbf{S}$ does not admit any
solution $u_{0}$ such that $J(u_{0}) =\alpha (\mathbf{S}%
) $. We need the following lemmas to show our main results.

\begin{lemma} \label{m1}
For each positive number $\varepsilon (\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $ and $l\geq 0$, there exists a $%
\delta (\varepsilon ,l) >0$ such that if $u\in \mathbf{M}%
_{0}(\mathbf{S}) $ and $J(u) \leq \alpha (\mathbf{A}^{r}) +\delta (\varepsilon ,l)$,
 then either $\int_{\mathbf{S}_{-l}^{+}}| u| ^{p}<\varepsilon $ or
 $\int_{\mathbf{S}_{l}^{-}}|u| ^{p}<\varepsilon $.
\end{lemma}

\begin{proof}
We divide the proof into the following steps:

\noindent Step 1:
Suppose that there exist $c>0,l_{0}\geq 0$ and
$\{u_n\} \subset \mathbf{M}_{0}(\mathbf{S}) $ such that
\begin{gather}
J(u_{n}) =\alpha (\mathbf{A}^{r}) +o(1) \,, \\
\int_{\mathbf{S}_{-l_{0}}^{+}}| u_{n}| ^{p}\geq c\,,  \label{27}\\
\int_{\mathbf{S}_{l_{0}}^{-}}| u_{n}| ^{p}\geq c\,.  \label{28}
\end{gather}
 From Lemma \ref{p7}, $\{u_n\} $ is a (PS)$_{\alpha (\mathbf{A}^{r})}$-sequence
 in $H_{0}^{1}(\mathbf{S}) $ for $J$. Since $\mathbf{S}$ is a proper large
domain in $\mathbf{A}^{r}$, by
Proposition \ref {c1} and Lemma \ref{e4}, there exists a
subsequence $\{u_n\} $ such that $\{ \xi_{n}u_{n}\} $ is also a
(PS)$_{\alpha (\mathbf{S}) }$-sequence in $H_{0}^{1}(\mathbf{S}) $ for
$J$,  where $\xi _{n}$ is as in (\ref{11}). Let $v_{n}=\xi _{n}u_{n}$, we
obtain
\begin{equation}
\begin{gathered}
J(v_{n}) = \alpha (\mathbf{A}^{r}) +o(1),  \\
J'(v_{n}) = o(1)\quad\text{in }H^{-1}(\mathbf{S}),
\end{gathered}  \label{29}
\end{equation}
and there exists a $n_{0}>l_{0}$ such that $v_{n}=0$ in $\overline{\mathbf{A}%
(n_{0}) }$ for $n>2n_{0}$, where $\mathbf{A}(n) =%
\mathbf{S}_{-n,n}$. Moreover, $v_{n}=v_{n}^{+}+v_{n}^{-}$ and
\[
v_{n}^{\pm }(z)=\begin{cases}
v_{n}(z) & \text{for }z\in \mathbf{S}_{\mp l_{0}}^{\pm }, \\
0 & \text{for }z\notin \mathbf{S}_{\mp l_{0}}^{\pm }.
\end{cases}
\]
Then $v_{n}^{\pm }\in H_{0}^{1}(\mathbf{S}_{\mp l_{0}}^{\pm })$
and $a(v_{n}^{\pm }) =b(v_{n}^{\pm }) +o(1) $. By (\ref{29}), we obtain
\[
J'(v_{n}^{\pm })=o(1)\quad \text{strongly in }H^{-1}(\mathbf{S}_{\mp
l_{0}}^{\pm }).
\]
Thus,
\[
\alpha (\mathbf{A}^{r}) +o(1) =J(v_{n})
=J(v_{n}^{+}) +J(v_{n}^{-}).
\]
Assume that
$J(v_{n}^{\pm })=c^{\pm }+o(1)$.
Then
\begin{equation}
c^{+}+c^{-}=\alpha (\mathbf{A}^{r}) .  \label{30}
\end{equation}
Since $c^{\pm }$ are (PS)-values in $H_{0}^{1}(\mathbf{S}_{\mp
l_{0}}^{\pm })$ for $J$, they
are nonnegative. Moreover, the half strips $\mathbf{S}_{-l_{0}}^{+}$ and
$\mathbf{S}_{l_{0}}^{-}$ are proper large domains in
$\mathbf{A}^{r}$,  From Lemma \ref{e4}, we have
\begin{equation}
\alpha (\mathbf{A}^{r})=\alpha (\mathbf{S}_{-l_{0}}^{+})=\alpha (\mathbf{S}%
_{l_{0}}^{-}).  \label{31}
\end{equation}
Thus, by $(\ref{30}) $, $(\ref{31}) $ and the
definition of Nehari minimization problem, we may assume that $c^{+}=\alpha (%
\mathbf{S}_{-l_{0}}^{+})=\alpha (\mathbf{A}^{r}) $ and $c^{-}=0$.
Next, for $n>2n_{0}$,
\begin{align*}
\int_{\mathbf{S}}|u_n| ^{p} &= \int_{\mathbf{S}}|v_n| ^{p}+o(1) \\
&= \int_{\mathbf{S}_{-l_{0}}^{+}}| v_{n}^{+}| ^{p}
  +\int_{\mathbf{S}_{l_{0}}^{-}}|u_n| ^{p}+o(1) .
\end{align*}
Thus,
\begin{align*}
\int_{\mathbf{S}_{l_{0}}^{-}}|u_n| ^{p}
&=\int_{\mathbf{S}}|u_n| ^{p}-\int_{\mathbf{S}_{-l_{0}}^{+}}|v_{n}^{+}| ^{p}+o(1) \\
&=(\frac{2p}{p-2}) \alpha (\mathbf{A}^{r})-(\frac{2p}{p-2}%
) \alpha (\mathbf{A}^{r}) +o(1) \\
&=o(1) ,
\end{align*}
which contradicts to $(\ref{28})$.

\noindent Step 2: Suppose that there exists a
$u_{0}\in \mathbf{M}_{0}(\mathbf{S}) $ with $J(u_{0}) <\alpha (\mathbf{A}^{r})
+\delta (\varepsilon ) $ such that
\[
\int_{\mathbf{S}_{-l_{0}}^{+}}| u_{0}| ^{p}<\varepsilon
\quad\text{and}\quad
\int_{\mathbf{S}_{l_{0}}^{-}}| u_{0}| ^{p}<\varepsilon .
\]
Then
\begin{align*}
\frac{2p}{(p-2) }\alpha (\mathbf{S})
&\leq \int_{\mathbf{S}}| u_{0}| ^{p}=\int_{\mathbf{S}_{l_{0}}^{+}}|
u_{0}| ^{p}+\int_{\mathbf{S}_{l_{0}}^{-}}| u_{0}| ^{p} \\
&< \frac{p}{(p-2) }\alpha (\mathbf{A}^{r}) +\frac{p}{%
(p-2) }\alpha (\mathbf{A}^{r}) \\
&= \frac{2p}{(p-2) }\alpha (\mathbf{A}^{r}) ,
\end{align*}
which is also a contradiction.
\end{proof}

\begin{lemma} \label{m2}
If $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r}) $.
Then for each $0<\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $,
there exist positive numbers $l(\varepsilon ) $ and $\delta (\varepsilon) $
such that if $u\in \mathbf{M}_{s}(\mathbf{S}) $ and
$J(u) <\alpha _{s}(\mathbf{S}) +\delta (\varepsilon ) $, then $\int_{(\mathbf{S}_{-l(\varepsilon ) ,l(\varepsilon) }) ^{c}}|u| ^{p}<\varepsilon $.
\end{lemma}

\begin{proof}
If not, there exist a positive number $c\leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $ and
$\{u_n\} \subset \mathbf{M}_{s}(\mathbf{S}) $ such that
\begin{equation}
\begin{gathered}
J(u_{n}) =\alpha _{s}(\mathbf{S}) +\frac{1}{n}\,,\\
\int_{(\mathbf{S}_{-n,n}) ^{c}}|u_n| ^{p}\geq c%
\quad\text{for all }n=1,2,\ldots .
\end{gathered} \label{16}
\end{equation}
By Lemma \ref{p7}, $\{u_n\} $ is a (PS)$_{\alpha
_{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S}) $
for $J$. Since $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A%
}^{r}) $. By Proposition \ref{e7}, $J$ is satisfying
(PS)$_{\alpha _{s}(\mathbf{S}) }-$condition in
$H_{s}(\mathbf{S}) $. Thus, there exist a subsequence
$\{u_n\} $ and $u_{0}\in H_{s}(\mathbf{S}) $ such that
\[
u_{n}\to u_{0}\quad\text{strongly in }H_{s}(\mathbf{S}) .
\]
By the Sobolev imbedding theorem and the Vitali convergence
theorem, there exists a $l_{0}>0$ such that
\[
\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|u_n| ^{p}<%
\frac{c}{2}\quad\text{for all }n,
\]
which contradicts to $(\ref{16}) $.
\end{proof}

\begin{lemma}\label{m3}
Suppose that the equation $(\ref{E1}) $ in
$\mathbf{S}$ does not admit any solution $u_{0}$ such that
$J(u_{0}) =\alpha _{s}(\mathbf{S}) $. Then
for each positive number $\varepsilon
\leq (\frac{2p}{p-2}) \alpha _{s}(\mathbf{S}) $ and $%
l$, there exists a $\delta (\varepsilon ,l) >0$ such
that if $u\in \mathbf{M}_{s}(\mathbf{S}) $ and
$J(u) <\alpha
_{s}(\mathbf{S}) +\delta (\varepsilon ,l) $, then $%
\int_{\mathbf{S}_{-l,l}}|u| ^{p}<\varepsilon $.
\end{lemma}

The proof of this lemma is similar to the proof of Lemma \ref{m1}, and
is omitted here.

 For $\Theta _{t}=\mathbf{A}_{-t,t}^{r}\backslash \overline{\omega }$,
 consider the filtration of $J$ in $\mathbf{M}(\Theta_{t}) $,
\[
F(\Theta _{t}) =\left\{ u\in \mathbf{M}_{0}(\Theta
_{t}) :J(u) \leq \alpha _{s}(\mathbf{S})
\right\} .
\]
Note that if $F(\Theta _{t}) $ is a nonempty set, then
\[
\alpha (\Theta _{t}) =\inf_{v\in F(\Theta _{t})
}J(v) .
\]
Note that $\Theta _{t_{1}}\subset \Theta _{t_{2}}$ for
$t_{1}<t_{2}$. Thus, $\alpha _{X}(\Theta _{t_{1}})
>\alpha _{X}(\Theta _{t_{2}}) $ for $t_{1}<t_{2}$.
Then we have the following result.

\begin{lemma} \label{m4}
$\alpha _{X}(\Theta _{t}) \searrow \alpha _{X}(\mathbf{S}) $ as $t\nearrow \infty $.
\end{lemma}

The proof of this lemma is similar to the proof of Lien-Tzeng-Wang
\cite[Lemma 2.5]{LTW} and omitted here.

\begin{theorem} \label{t1}
There exists a positive number $t_{0}$ such that $F(\Theta
_{t}) $ is non-empty and $F(\Theta _{t}) \cap \mathbf{M}%
_{s}(\Theta _{t}) =\phi $ for $t\geq t_{0}$.
Furthermore, the equation $(\ref{E1}) $ in $\Theta
_{t}$ has three positive solutions which one is $y$-symmetric and
the other are non-axially symmetric for $t\geq t_{0}$.
\end{theorem}

\begin{proof}
First, we need to show that $\alpha (\mathbf{S})<\alpha _{s}(\mathbf{S}) $.
Assume the contrary, $\alpha (\mathbf{S}) =\alpha _{s}(\mathbf{S}) $.
By Corollary \ref {e9}, the equation $(\ref{E1}) $ in $\mathbf{S}$ admits a
solution $u_{0}$ such that $J(u_{0}) =\alpha (\mathbf{S})$, this contradicts the fact of Lemma \ref{e4}. Since $\mathbf{S}$
is a proper large domain in $\mathbf{A}^{r}$. From Lemma \ref{e4}, we have
\begin{equation}
\alpha (\mathbf{A}^{r}) =\alpha (\mathbf{S}) <\alpha
_{s}(\mathbf{S}) .  \label{18}
\end{equation}
By $(\ref{18}) $ and Lemma \ref{m4}, there exists a $t_{0}>0$
such that
\begin{equation}
\alpha (\mathbf{S}) <\alpha (\Theta _{t}) \leq
\alpha _{s}(\mathbf{S}) \quad\text{for all }t\geq t_{0}.  \label{20}
\end{equation}
Since $\Theta_{t}$ is a $y$-symmetric bounded domain, by Lemma
\ref{c8},  $F(\Theta _{t}) $ is nonempty for
all $t\geq t_{0}$. Moreover,
\[
\alpha _{s}(\Theta _{t}) =\inf_{v\in \mathbf{M}_{s}(\Theta
_{t}) }J(v)
\]
and
\begin{equation}
\alpha _{s}(\mathbf{S}) <\alpha _{s}(\Theta _{t})
\quad\text{for all }t>0.  \label{21}
\end{equation}
We can conclude that $F(\Theta _{t}) \cap
\mathbf{M}_{s}(\Theta _{t}) =\phi $ for all $t\geq
t_{0}$. By $(\ref{20}) $, $(\ref{21}) $
and Lemma \ref{c8}, we have
\begin{equation}
\alpha (\Theta _{t}) \leq \alpha _{s}(\mathbf{S}) <\alpha _{s}(\Theta _{t}) \quad\text{for
 all } t\geq t_{0} \label{22}
\end{equation}
and the equation $(\ref{E1}) $ in $\Theta _{t}$ admit two
disjoint positive solutions $u_{1},u_{2}$ such that $J(u_{1})
=\alpha _{s}(\Theta _{t}) $ and $J(u_{2}) =\alpha
(\Theta _{t}) $. Take $u_{3}(x,y) =u_{2}(x,-y)$, then $J(u_{3}) =\alpha (\Theta _{t}) $%
, $u_{3}\in \mathbf{M}_{0}(\Theta _{t}) $ and $u_{3}$ is third
positive solution.
\end{proof}

\begin{remark} \rm
By Theorem \ref{t1}, there exists a $t_{0}>0$ such that for $t\geq
t_{0}$, the equation $(\ref{E1}) $ in $\Theta _{t}$ has one $y-$%
symmetric positive solution $u_{1}$ and two non-axially symmetric positive
solutions $u_{2}$ and $u_{3}$. Moreover,
\[
\int_{\Theta _{t}}|u_1| ^{p}=\frac{2p}{p-2}\alpha _{s}(\Theta _{t}) >\frac{2p}{p-2}\alpha _{s}(\mathbf{S})
\]
and
\[
\int_{\Theta _{t}}| u_{i}| ^{p}=\frac{2p}{p-2}\alpha (\Theta _{t}) \leq \frac{2p}{p-2}\alpha _{s}(\mathbf{S})
\quad\text{for }i=2,3.
\]
Thus, we can conclude that
\begin{gather*}
\int_{\Theta _{t}^{+}}|u_1| ^{p}=\int_{\Theta _{t}^{-}}|u_1| ^{p}>\frac{p}{p-2}\alpha _{s}(\mathbf{S}) ,\\
\int_{\Theta _{t}^{+}}|u_2| ^{p}\leq \frac{p}{p-2}\alpha
_{s}(\mathbf{S})\\
\int_{\Theta _{t}^{-}}|u_3| ^{p}\leq \frac{p}{p-2}\alpha
_{s}(\mathbf{S}) ,
\end{gather*}
where $\Theta _{t}^{+}=\left\{ (x,y) \in \Theta _{t}:y\geq
0\right\} $ and $\Theta _{t}^{-}=\left\{ (x,y) \in \Theta
_{t}:y\leq 0\right\} $.
\end{remark}

 Next, we describe the concentration of solutions of equation $%
(\ref{E1}) $ in $\Theta _{t}$. We need the following notation:
\begin{gather*}
\Theta _{t}(-l,l) =\left\{ (x,y) \in \Theta_{t}:-l\leq y\leq l\right\} ; \\
\Theta _{t}^{+}(l) =\left\{ (x,y) \in \Theta_{t}:y\geq l\right\} ; \\
\Theta _{t}^{-}(l) =\left\{ (x,y) \in \Theta_{t}:y\leq l\right\} .
\end{gather*}
Then we have the following results.

\begin{theorem} \label{t2}
Suppose that $\alpha _{s}(\mathbf{S})<2\alpha (\mathbf{A}^{r}) $.
Then for each positive number
$\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $, there
exist positive numbers $t_{0}>l_{0}$ such that for $t>t_{0}$ the equation $%
(\ref{E1}) $ in $\Theta _{t}$ has three positive solutions $u_{1}$, $u_{2}$ and
$u_{3}$. Moreover,\begin{itemize}
\item[(i)] $\int_{(\Theta _{t}(-l_{0},l_{0}) )^{c}}|u_1| ^{p}<\varepsilon $
\item[(ii)] $\int_{\Theta _{t}^{+}(-l_{0}) }|u_2| ^{p}<\varepsilon $ and
$\int_{\Theta _{t}^{-}(l_{0}) }|u_3| ^{p}<\varepsilon $.
\end{itemize}
\end{theorem}

\begin{proof}
Since $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r}) $. By
Lemma \ref{m2}, for each positive number $\varepsilon \leq
(\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $, there exist positive
numbers $l_{0}$ and $\delta (\varepsilon ) $ such that if $u\in
\mathbf{M}_{s}(\mathbf{S}) $ and $J(u) <\alpha _{s}(\mathbf{S})
+\delta (\varepsilon ) $, then $\int_{(\mathbf{S}_{-l_{0},l_{0}})
^{c}}| u| ^{p}<\varepsilon $. Moreover, by Lemma \ref{m4}, there
exists a $t_{1}>0$
such that $\alpha _{s}(\Theta _{t}) <\alpha _{s}(\mathbf{S}%
) +\delta (\varepsilon ) $ for all $t>t_{1}$. Since $%
\Theta _{t}$ is a bounded domain, by Lemma \ref{c8}, the equation $(\ref{E1}) $ in $\Theta _{t}$ admits a positive solution $u_{1}\in
H_{0}^{1}(\Theta _{t}) $ such that $J(u_{1}) =\alpha
_{s}(\Theta _{t}) $. Thus, $u_{1}\in \mathbf{M}_{s}(\mathbf{S})$,
\begin{gather*}
J(u_{1}) <\alpha (\mathbf{A}^{r}) +\delta (\varepsilon )\,,\\
\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|u_1| ^{p}
=\int_{(\Theta _{t}(-l_{0},l_{0}) ) ^{c}}|u_1|^{p}<\varepsilon .
\end{gather*}
Fixed the positive numbers $\varepsilon ,l_{0}$. By Lemma \ref{m1}, there exists
a $\delta (\varepsilon ,l_{0}) >0$ such that if $u\in \mathbf{M}
_{0}(\mathbf{S}) $ and $J(u) <\alpha (\mathbf{A
}^{r}) +\delta (\varepsilon ,l_{0}) $, then $\int_{\mathbf{S
}_{-l_{0}}^{+}}|u| ^{p}<\varepsilon $ or $\int_{\mathbf{S}
_{l_{0}}^{-}}|u| ^{p}<\varepsilon $. Moreover, by Lemma \ref{m4},
there exists a $t_{2}>0$ such that $\alpha (\Theta _{t})
<\alpha (\mathbf{A}^{r}) +\delta (\varepsilon ) $
for all $t>t_{2}$. Since $\Theta _{t}$ is a bounded domain, by Lemma \ref{c8},
the equation $(\ref{E1}) $ in $\Theta _{t}$ admits a
positive solution $u_{2}$ such that $J(u_{2}) =\alpha
(\Theta _{t}) $. Then $u_{2}\in \mathbf{M}_{0}(\Theta _{t}) \subset \mathbf{M}_{0}(\mathbf{S})
,\;J(u_{2}) <\alpha (\mathbf{A}^{r})
+\delta (\varepsilon ) $ and either
\begin{equation}
\int_{\Theta _{t}^{+}(-l_{0}) }|u_2|
^{p}<\varepsilon \;\text{or\ }\int_{\Theta _{t}^{-}(l_{0})
}|u_2| ^{p}<\varepsilon .  \label{23}
\end{equation}
Without loss of generality, we may assume that
\[
\int_{\Theta _{t}^{+}(-l_{0}) }|u_2|^{p}<\varepsilon .
\]
Take $u_{3}(x,y) =u_{2}(x,-y)$, then $u_{3}$ is
third positive solution and
\[
\int_{\Theta _{t}^{-}(l_{0}) }|u_3|
^{p}<\varepsilon .
\]
Now, let $t_{0}=\max \{ t_{1},t_{2}\} $. Since
$\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $, $u_{i}$
is disjoint for $i=1,2,3$.
\end{proof}

\begin{theorem}\label{t3}
Suppose that the equation $(\ref{E1}) $ in $\mathbf{S}$ does not
admit any solution $u_{0}$ such that $J(u_{0}) =\alpha
_{s}(\mathbf{S}) $. Then for positive numbers $\varepsilon \leq
(\frac{p}{p-2}) \alpha (\mathbf{A}^{r}) $ and $l $, there exists a
positive number $t_{0}$ such that for $t>t_{0}$, the equation
$(\ref{E1}) $ in $\Theta _{t}$ has three positive solutions
$u_{1}, u_{2}$ and $u_{3}$. Moreover, \begin{itemize}
\item[(i)] $\int_{(\Theta_{t}(-l,l) ) ^{c}}|u_1|^{p}<\varepsilon $
\item[(ii)] $\int_{\Theta_{t}^{+}(-l) }|u_2| ^{p}<\varepsilon $
and $\int_{\Theta _{t}^{-}(l) }|u_3|^{p}<\varepsilon $.
\end{itemize}
\end{theorem}

The proof of this theorem is similar to the proof of Theorem \ref{t2} and
therefore omitted here.

Note that if $u_{1},u_{2}$ and $u_{3}$ are positive solutions as in
Theorem \ref{t2} or Theorem \ref{t3}, then $u_{1}$ is $y$-symmetric
and $u_{2},u_{3}$ are non-axially symmetric.

\section{Dynamic System of Solutions}

For $m=1,2,\cdots $, define $\Theta
_{m}=\mathbf{A}_{-m,m}^{r}\backslash \overline{\omega }$, then
$\{\Theta _{m}\}$ is an increasing sequence and
\[
\mathbf{S}=\mathbf{A}^{r}\backslash \overline{\omega }
=\cup_{m=1}^\infty \Theta _{m}.
\]
By Theorem \ref{t1}, there exists a $t_{0}>0$ such that for $m\geq
t_{0}$,
the equation $(\ref{E1}) $ in $\Theta _{m}$ admit one $y-$%
symmetric positive solution $u_{m}^{1}$ and two non-axially
symmetric positive solutions $u_{m}^{2}$ and $u_{m}^{3}$. Note
that
\[
J(u_{m}^{2}) =J(u_{m}^{3}) =\alpha (\Theta
_{m}) <\alpha _{s}(\Theta _{m}) =J(u_{m}^{1})
\quad\text{for all }m\geq t_{0}.
\]
 Then we have the following results.

\begin{theorem} \label{t4} \begin{itemize}
\item[(i)] The sequence $\{ u_{m}^{1}\} $ is a
(PS)$_{\alpha _{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S}) $ for $J$

\item[(ii)] If $\alpha _{s}(\Theta _{m_{0}}) <2\alpha(\mathbf{A}^{r}) $ for
some $m_{0}>0$, then there exist a subsequence $u_{m}^{1}$ and
$u^{1}\in H_{s}(\Omega) $ such that $u_{m}^{1}\to u^{1}$
strongly in $L^{p}(\mathbf{S} )$ in $H_{s}(\mathbf{S} ) $ as
$m\to \infty $ and $J(u^{1}) =\alpha _{s}(\mathbf{S} )$

\item[(iii)] If the equation $(\ref{E1}) $ in $\mathbf{S%
}$ does not admit any solution $u_{0}$ such that $J(u_{0})
=\alpha _{s}(\mathbf{S})$, then $u_{m}^{1}\rightharpoonup 0$
weakly in $L^{p}(\mathbf{S}) $ and in $H_{0}^{1}(\mathbf{S}) $ as
$m\to \infty $.
\end{itemize}
\end{theorem}

\begin{proof}
(i) By Lemma \ref{m4}, we have $J(u_{m}^{1}) =\alpha _{s}(\Theta _{m}) =\alpha
_{s}(\mathbf{S}) +o(1) $. Since
$u_{m}^{1}\in \mathbf{M}_{s}(\Theta _{m}) \subset
\mathbf{M}_{s}(\mathbf{S})$, from Lemma \ref {p7} we
can conclude that $\{ u_{m}^{1}\} $ is a (PS)$_{\alpha
_{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S}) $ for $J$.

\noindent (ii) Since
$\alpha _{s}(\Theta _{m_{0}}) <2\alpha (\mathbf{A}^{r}) $ for some $m_{0}>0$ and $\Theta _{m}\subset
\Theta _{m+1}\subset \mathbf{S}$ for each $m$, we have $\alpha
_{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r})$. By
Proposition \ref{e7}, $J$ satisfies the
(PS)$_{\alpha _{s}(\mathbf{S})}-$condition in $H_{s}(\mathbf{S})$.
Then there exist a subsequence
$\left\{ u_{m}^{1}\right\} $ and a $y$-symmetric positive solution
$u^{1}$ of equation $(\ref{E1}) $ in $\mathbf{S}$ such that
$u_{m}^{1}\to u^{1}$ strongly in $L^{p}(\mathbf{S})$ and in
$H_{s}(\mathbf{S}) $ and $J(u^{1}) =\alpha
_{s}(\mathbf{S}) $.

\noindent (iii) Let $v\in L^{q}(\mathbf{S}) $, where $\frac{1}{p}+\frac{1}{q}=1$.
Then for each $\varepsilon >0$, there exists a $l>0$ such that
\[
\int_{(\mathbf{S}_{-l,l}) ^{c}}| v| ^{q}<\varepsilon ^{q}.
\]
Moreover, by Theorem \ref{t3}, there exists a $m_{0}$ such that
\[
\int_{\mathbf{S}_{-l,l}}| u_{m}^{1}| ^{q}<\varepsilon ^{p}
\quad\text{for all }m>m_{0}.
\]
Thus, for each $\varepsilon >0$, there exists a $m_{0}$ such that
\begin{align*}
\int_{\mathbf{S}}u_{m}^{1}v
&= \int_{(\mathbf{S}_{-l,l})^{c}}u_{m}^{1}v+\int_{\mathbf{S}_{-l,l}}u_{m}^{1}v \\
&\leq \Big(\int_{(\mathbf{S}_{-l,l}) ^{c}}|u_{m}^{1}| ^{p}\Big)
^{1/p}
\Big(\int_{(\mathbf{S}_{-l,l}) ^{c}}| v| ^{q}\Big) ^{\frac{1}{q}}
+\Big(\int_{\mathbf{S}_{-l,l}}| u_{m}^{1}| ^{p}\Big) ^{1/p}
\Big(\int_{\mathbf{S}_{-l,l}}| v| ^{q}\Big) ^{\frac{1}{q}} \\
&\leq (c_{1}+c_{2}) \varepsilon \quad\text{for all }m>m_{0},
\end{align*}
where $c_{1}=(\frac{2p}{p-2}\alpha _{s}(\Theta _{1})
) $ and $c_{2}=\left\| v\right\| _{L^{q}}$. This implies
$u_{m}^{1}\rightharpoonup 0$ weakly in $L^{p}(\mathbf{S}) $ as
$m\to \infty $. Since $u_{m}^{1}$ is a solution of equation
$(\ref{E1}) $ in $\Theta _{m}$, we have
\[
\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi +u_{m}^{1}\varphi
=\int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \quad\text{for
all }\varphi \in H_{0}^{1}(\Theta _{m}) .
\]
First, we need to show for each $\varepsilon >0$ and $\varphi \in
C_{c}^{1}(\mathbf{S}) $, there exists $m_{0}$ such
that
\[
\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi +u_{m}^{1}\varphi
<\varepsilon \quad\text{for all }m>m_{0}.
\]
For $\varphi \in C_{c}^{1}(\mathbf{S}) $. Let $K=\mathop{\rm supp}\varphi $,
then $K\subset \mathbf{S}$ is compact and there exists a $m_{1}$
such that $K\subset \Theta _{m}$ for all $m\geq m_{1}$. Thus, by Theorem \ref
{t3} for each $\varepsilon >0$, there exist $l_{0}>0$ and $m_{0}$ such that $%
\varphi \in H_{0}^{1}(\Theta _{m}) $,
\begin{gather*}
\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|\varphi|^{p}=0\,,\\
\int_{\mathbf{S}_{-l_{0},l_{0}}}| u_{m}^{1}| ^{p}
<\varepsilon ^{\frac{p-1}{p}}\quad\text{for all }m>m_{0}.
\end{gather*}
We obtain
\begin{align*}
\int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi
&=\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|
u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi +\int_{\mathbf{S}_{-l,l_{0}}}|
u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \\
&\leq \Big(\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|
u_{m}^{1}| ^{p}\Big) ^{\frac{p-1}{p}}\Big(\int_{(\mathbf{S}
_{-l_{0},l_{0}}) ^{c}}| \varphi | ^{p}\Big) ^{1/p}
\\
&\quad+\Big(\int_{\mathbf{S}_{-l_{0},l_{0}}}| u_{m}^{1}|
^{p}\Big) ^{\frac{p-1}{p}}\Big(\int_{\mathbf{S}_{-l_{0},l_{0}}}|
\varphi | ^{p}\Big) ^{1/p}
\leq c\varepsilon
\end{align*}
and
\begin{align*}
\int_{\mathbf{S}}\nabla u_{m}^{1}\nabla \varphi +\int_{\mathbf{S}%
}u_{m}^{1}\varphi &=\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi
+\int_{\Theta _{m}}u_{m}^{1}\varphi \\
&=\int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \quad\text{for all }m>m_{0}.
\end{align*}
This follows that
\begin{equation}
\int_{\mathbf{S}}\nabla u_{m}^{1}\nabla \varphi +\int_{\mathbf{S}%
}u_{m}^{1}\varphi \leq c\varepsilon \quad\text{for all }m>m_{0}.  \label{36}
\end{equation}
Since $\alpha _{s}(\Theta _{m+1}) <\alpha _{s}(\Theta )$, there exists a $C>0$ such that $\Vert
u_{m}^{1}\Vert _{H^{1}}\leq C$. Thus, for each $\varepsilon >0$
and $\psi \in H_{0}^{1}(\mathbf{S})$, there exists a $\varphi \in
C_{c}^{1}(\mathbf{S}) $ such that
\begin{equation}
\Vert \psi -\varphi \Vert _{H^{1}}<\frac{\varepsilon }{C}.  \label{37}
\end{equation}
From $(\ref{36}) $ and $(\ref{37})$, we
can conclude that for each $\varepsilon >0$ and $\psi \in
H_{0}^{1}(\mathbf{S})$, there exists a $m_{0}>0$ such that
\begin{align*}
\left\langle u_{m}^{1},\psi \right\rangle _{H^{1}}
&=\left\langle u_{m}^{1},\psi -\varphi \right\rangle _{H^{1}}
+\left\langle u_{m}^{1},\varphi \right\rangle _{H^{1}} \\
&\leq C\Vert \psi -\varphi \Vert _{H^{1}}+\left\langle u_{m}^{1},\varphi
\right\rangle _{H^{1}} \\
&< \varepsilon +c\varepsilon \quad\text{for }m>m_{0}.
\end{align*}
This implies $u_{m}^{1}\rightharpoonup 0$ weakly in $H_{0}^{1}(\mathbf{S}) $.
\end{proof}

\begin{theorem} \label{t5} \begin{itemize}
\item[(i)]  The sequence $\{ u_{n}^{i}\} $ is a
(PS)$_{\alpha (\mathbf{S}) }$-sequence in $H_{0}^{1}(\mathbf{S}) $ for $J$,
for $i=2,3$
\item[(ii)] $u_{n}^{i}\rightharpoonup 0$ weakly in $L^{p}(\mathbf{S}) $ and
in $H_{0}^{1}(\mathbf{S}) $ as $n\to \infty$, for $i=2,3$.
\end{itemize}
\end{theorem}

The proof of this theorem is similar to the proof of Theorem \ref{t4} (i) and (iii).

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\end{document}