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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 291, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/291\hfil Multiple positive solutions]
{Multiple positive solutions for a \\ Kirchhoff type problem}

\author[Wenli Zhang\hfil EJDE-2015/291\hfilneg]
{Wenli Zhang}

\address{Wenli Zhang \newline
Department of Mathematics,
Changzhi University, Shanxi 046011, China}
\email{ywl6133@126.com}

\thanks{Submitted September 2, 2014. Published November 23, 2015.}
\subjclass[2010]{35J50}
\keywords{Positive solutions; Kirchhoff type equation; Morse theory;
\hfill\break\indent Ljusternik-Schnirelmann category}

\begin{abstract}
 In this article, we study the existence and multiplicity of positive 
 solutions  of a Kirchhoff type equation on a smooth bounded domain 
 $\Omega\subset \mathbb{R}^3$,  and we show that the number of positive 
 solutions of the equation depends  on the topological properties of 
 the domain. The technique is based on  Ljusternik-Schnirelmann category 
 and Morse theory.
\end{abstract}

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

\section{Introduction}

This article concerns the multiplicity of positive solutions to the
elliptic problem
\begin{equation}
\begin{gathered}
  -\Big(\varepsilon^2a+\varepsilon b\int_{\Omega}|\nabla u|^2\Big)
 \Delta u+u=|u|^{p-1}u,\quad x\in \Omega \\
   u=0,\quad x\in \partial\Omega
\end{gathered} \label{ePe}
\end{equation}
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^3$, $\varepsilon>0$,
 $a,b>0$ are constants, $3<p<5$.

In recent years, some mathematicians considered the  problem
\begin{equation}
 \begin{gathered}
  -\Big(a+b\int_{\Omega}|\nabla u|^2\Big)\Delta u=f(x,u),\quad
x\in \Omega \\
    u=0,\quad x\in\partial\Omega
    \end{gathered} \label{e1.1}
\end{equation}
where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain.
See for example\cite{Al,Ch1,Ch2,He1,Ma,Pe,Ya,Zh}.

When $a=1$, $b=0$, $\mathbb{R}^3$ is replaced by $\mathbb{R}^N$, and
$|u|^{p-1}u$ is replaced by  $f(u)$, Equation \eqref{ePe}  reduces to
\begin{equation}
\begin{gathered}
  -\varepsilon^2\bigtriangleup u+u=f(u),\quad x\in \Omega \\
    u=0,  \quad x\in \partial\Omega
    \end{gathered} \label{e1.2}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^N$.
Benci and Cerami \cite{Be2} used Morse theory to estimate the number
of positive solutions of the problem \eqref{e1.2}.
They proved that for $\varepsilon$ sufficiently small the number of positive
solutions depends on the topology of $\Omega$, actually on
the Poincar\'e polynomial of $\Omega$, $P_t(\Omega)$, defined below.
Candela and Lazzo \cite{Ca}
considered the same equation with mixed Dirichlet-Neumann boundary conditions
and $f(t)=|t|^{p-2}t$.
It was proved that the number of positive solutions is influenced by the topology
of the part $\Gamma_1$ of the boundary $\partial\Omega$ where $\varepsilon$
is sufficiently small. Recently, Benci, Bonanno, Ghimenti and Micheletti
\cite{Be1,Gh2} proved that the number of solutions of  \eqref{e1.1}  with
$f(t)=|t|^{p-2}t$ on a smooth bounded domain $\Omega\subset \mathbb{R}^3$
depends on the topological properties of the domain.
More recently, Ghimenti and Micheletti \cite{Gh1}
extended the result in \cite{Be1,Gh2} to the Klein-Gordon-Maxwell and
Schr\"oedinger-Maxwell system and showed that the geometry of the
3-dimensional Riemannian manifold has effects on the number of solutions
of both systems.

Moreover, as far as we known, the existence and multiplicity of nontrivial
solutions to the Kirchhoff equation have not ever been studied by using Morse theory.
Motivated by the works described above and the fact, we will try to get the
multiplicity of positive solutions to \eqref{ePe} by using Ljusternik-Schnirelmann
category and Morse theory. So in this paper we shall fill this gap.

Our main results read as follows.

\begin{theorem} \label{thm1.1}
 Let $3<p<5$. For $\varepsilon>0$ sufficiently small, the problem \eqref{ePe}
has at least $\operatorname{cat}(\Omega)$ positive solutions.
\end{theorem}

\begin{theorem} \label{thm1.2}
Let $3<p<5$. Assume that for $\varepsilon>0$ sufficiently small all the
solutions of the problem \eqref{ePe} are nondegenerate.
Then there are at least $2P_1(\Omega)-1$ positive solutions,
\end{theorem}

\section{Notation and preliminary results}

Throughout this article, we use the following norms for $u\in H_0^1(\Omega)$:
\begin{gather*}
\|u\|_\varepsilon=\Big(\frac{1}{\varepsilon^3}
\int_\Omega \varepsilon^2|\nabla u|^2dx\Big)^{1/2}, \quad
|u|_{\varepsilon,p}=\Big(\frac{1}{\varepsilon^3}\int |u|^pdx\Big)^{1/p}, \\
\|u\|=\Big(\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)^{1/2},\quad
|u|_p=\Big(\int_{\mathbb{R}^3}|u|^pdx\Big)^{1/p}
\end{gather*}
and we denote by $H_\varepsilon$ the Hilbert space $H_0^1(\Omega)$
endowed with $\|\cdot\|_\varepsilon$ norm.

Following the work by He and Zou \cite{He2}, we let  $U(x)$ be the positive
ground state solution of
\begin{equation}
\begin{gathered}
  -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\Big)\Delta u+u=|u|^{p-1}u,
\quad x\in \mathbb{R}^3 \\
   u\in H^1(\mathbb{R}^3),u(x)>0, \quad x\in \mathbb{R}^3
    \end{gathered} \label{ePi}
\end{equation}
and $I_\infty(U)=m_\infty=\inf_{u\in M_\infty} I_\infty(u)$, where
\begin{gather*}
I_\infty(u)=\frac{a}{2}\|u\|^2+\frac{1}{2}|u|_2^2+\frac{b}{4}\|u\|^4
-\frac{1}{p+1}|u^+|_{p+1}^{p+1}.
\\
M_\infty=\{u\in H^1(\mathbb{R}^3)\backslash \{0\}:G_\infty(u)
=\langle I'_\infty(u),u\rangle=0\}
\end{gather*}
For $\varepsilon>0$ we set $U_\varepsilon(x)=U(x/\varepsilon)$.
Obviously $U_\varepsilon(x)$ is the solution of the  problem
\begin{gather*}
  -\Big(\varepsilon^2a+\varepsilon b\int_{\Omega}|\nabla u|^2\Big)\Delta u+u
=|u|^{p-1}u,\quad  x\in \Omega \\
u\in H^1_0(\Omega),\quad u>0, \quad x\in \Omega
    \end{gather*}
Now we shall recall some topological tools.

\begin{definition}[\cite{Lj}] \label{def2.1} \rm
Let $X$ a topological space and consider a closed subset $A\subset X$.
We say that $A$ has category $k$ relative to $X(\operatorname{cat}_X(A)=k)$
if $A$ is covered by $k$ closed sets $A_i,i=1,2,\dots,k$, which are contractible
in $X$, and $k$ is the minimum integer with this property.
 We simply denote $\operatorname{cat}(X)=\operatorname{cat}_X(X)$.
\end{definition}

\begin{remark}[\cite{Be3}] \label{rmk2.2}  \rm
Let $X_1$ and $X_2$ be topological space. If $g_1:X_1\to X_2$ and
$g_2:X_2\to X_1$ are continuous operators such that $g_2\circ g_1$ is
homotopic to the identity on $X_1$, then
$\operatorname{cat}(X_1)\leq \operatorname{cat}(X_2)$.
\end{remark}

\begin{definition} \label{def2.3} \rm
Let $X$ is a topological space and let $H_k(X)$ denotes its $k$-th
 homology group with coefficients in $Q$. The Poincar\'e  polynomial
$P_t(X)$ of $X$ is defined as the following power series in $t$,
$$
P_t(X)=\sum_{k\geq 0}(\dim H_k(X))t^k\,.
$$
If $X$ is a compact space, we have that $\dim H_k(X)<\infty$
and this series is finite. In the case $P_t(X)$ is a polynomial and not
a formal series.
\end{definition}

\begin{remark}[\cite{Be2}] \label{rmk2.4} \rm
Let $X$ and $Y$ be topological spaces. If $f:X\to Y$ and $g:Y\to X$
are continuous operators such that $g\circ f$ is homotopic to the identity
on $X$, then $P_t(Y)=P_t(X)+Z(t)$ where $Z(t)$ is a polynomial with
nonnegative coefficients.
\end{remark}

\section{Proof of main results}

To prove our main results, we  consider the  functional
$I_\varepsilon\in C^2(H_\varepsilon,R)$, defined by
$$
I_\varepsilon (u)=\frac{a}{2}\|u\|_\varepsilon^2
+\frac{1}{2}|u|_{\varepsilon,2}^2+\frac{b}{4}\|u\|_\varepsilon^4
-\frac{1}{p+1}|u^+|_{\varepsilon,p+1}^{p+1}
$$
Obviously, there exists a one to one correspondence between the nontrivial
solutions of problem \eqref{ePe} and the nonzero critical points of
$I_\varepsilon$ on $H_\varepsilon$.

As the functional $I_\varepsilon$ is not bounded below on $H_\varepsilon$,
we introduce the manifold
$$
M_\varepsilon=\{u\in H_\varepsilon\backslash\{0\}:G_\varepsilon(u)
=\langle I'_\varepsilon(u),u\rangle=0\}
$$
Next, we present some properties of $I_\varepsilon$ and $M_\varepsilon$.

\begin{lemma} \label{lem3.1}
    (1) For any $u\in H_\varepsilon\backslash\{0\}$, there is a unique
$t_\varepsilon>0$ such that $u_{t_\varepsilon}(x)=t_\varepsilon u(x)
\in M_\varepsilon$.

(2) For any $\varepsilon>0$, $M_\varepsilon$ is a $C^1$ submanifold of
$H_\varepsilon$, and there exists $\sigma_\varepsilon>0$ and $K_\varepsilon>0$
such that for any $u\in M_\varepsilon$
$$
\|u\|_\varepsilon\geq \sigma_\varepsilon,\quad I_\varepsilon(u)\geq K_\varepsilon.
$$

(3) It holds $(PS)$ condition for the functional $I_\varepsilon$ on $M_\varepsilon$.
\end{lemma}

\begin{proof}
 (1) For any $u\in H_\varepsilon\backslash\{0\}$ and $t>0$, set $u_t(x)=tu(x)$.
Consider
$$
\Upsilon_\varepsilon(t)=I_\varepsilon(u_t)=\frac{a}{2}t^2\|u\|_\varepsilon^2
+\frac{1}{2}t^2|u|_{\varepsilon,2}^2
+\frac{b}{4}t^4\|u\|_\varepsilon^4-\frac{1}{p+1}t^{p+1}|u^+|_{\varepsilon,p+1}^{p+1}.
$$
By computing, we known that $\Upsilon_\varepsilon$ has a unique critical point
$t_\varepsilon>0$ corresponding to its maximum.
Then $\Upsilon_\varepsilon(t_\varepsilon)=\max_{t>0}\Upsilon_\varepsilon(t)$ and
$\Upsilon_\varepsilon'(t_\varepsilon)=0$. So
 $G_\varepsilon(u_{t_\varepsilon})=0$ and $u_{t_\varepsilon}\in M_\varepsilon$.

(2) By lemma \ref{lem3.1} (1), $M_\varepsilon\neq \emptyset$.
If $u\in M_\varepsilon$, using that $G_\varepsilon(u)=0$ and $3<p<5$, we have
$$
\langle G_\varepsilon'(u),u\rangle=a(2-(p+1))\|u\|_\varepsilon^2+(2-(p+1))|u|_{\varepsilon,2}^2+b(4-(p+1))\|u\|_\varepsilon^4<0. $$
So $M_\varepsilon$ is a $C^1$ manifold.
Using that $G_\varepsilon(u)=0$ and the Sobolev embedding,
we have
\begin{gather*}
a\|u\|_\varepsilon^2+|u|_{\varepsilon,2}^2+b
\|u\|_\varepsilon^4=|u^+|_{\varepsilon,p+1}^{p+1}\leq C\|u\|_\varepsilon^{p+1},\\
a\leq C\|u\|_\varepsilon^{p-1}.
\end{gather*}
So the conclusion $\|u\|_\varepsilon\geq \sigma_\varepsilon$ follows.
For any $u\in M_\varepsilon$,
$$
I_\varepsilon(u)=a(\frac{1}{2}-\frac{1}{p+1})\|u\|_\varepsilon^2
+(\frac{1}{2}-\frac{1}{p+1})|u|_{\varepsilon,2}^2
+b(\frac{1}{4}-\frac{1}{p+1})\|u\|_\varepsilon^4.
$$
Using $3<p<5$ and $\|u\|_\varepsilon\geq \sigma_\varepsilon$,
the conclusion $I_\varepsilon(u)\geq K_\varepsilon$ follows.

(3) Let $\{u_n\}$ is $(PS)$ sequence for $I_\varepsilon$ on $M_\varepsilon$, that is
$$
I_\varepsilon(u_n)\to c,~~~I'_\varepsilon|_{M_\varepsilon}(u_n)\to 0.
$$
Then it is easy to prove that $\|u_n\|_\varepsilon$ is bounded.
Going if necessary to a subsequence, we can assume that
$\|u_n\|_\varepsilon^2\to A(>0)$,
$u_n\rightharpoonup u$ in $H_\varepsilon$,
$u_n\to u$ in $L^s(\Omega)$ $(1\leq s<6)$.
Obviously, we have
$$
\rho_\varepsilon(u)=a\|u\|_\varepsilon^2+|u|_{\varepsilon,2}^2
+bA\|u\|_\varepsilon^2-|u^+|_{\varepsilon,p+1}^{p+1}=0.
$$
Set $\omega_n=u_n-u$. By Br\'ezis-Lieb Lemma, we have
$\|\omega_n\|_\varepsilon^2=\|u_n\|_\varepsilon^2-\|u\|_\varepsilon^2+o_n(1)$.
Since $\langle I'_\varepsilon(u_n),u_n\rangle=o_n(1)$, we obtain
$$
(a+bA)\|\omega_n\|_\varepsilon^2+\rho_\varepsilon(u)=o_n(1).
$$
This concludes the proof.
\end{proof}

By Lemma \ref{lem3.1} (2), we obtain $I_\varepsilon|_{M_\varepsilon}$
is bounded from below. By using Lagrange multiplier method, we known
that $M_\varepsilon$ contains every nonzero solution of problem \eqref{ePe},
and define the minimax $m_\varepsilon$ as
$$
m_\varepsilon=\inf_{u\in M_\varepsilon}I_\varepsilon(u)
$$


\begin{proof}[Proof of Theorem \ref{thm1.1}]
Since the functional $I_\varepsilon\in C^2$ is bounded below and satisfies the
(PS) condition on the complete  manifold $M_\varepsilon$, we have,
by the classical Ljusternik-Schnirelmann category result  \cite{Be5},
that $I_\varepsilon$ has at least $cat I_\varepsilon^d$ critical points
in the sublevel
$$
I_\varepsilon^d=\{u\in H_\varepsilon:I_\varepsilon(u)\leq d\}
$$
In the following, we will prove that, for $\varepsilon$ and $\delta$ sufficiently
small, it holds
$$
\operatorname{cat}(\Omega)
\leq \operatorname{cat} (M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta})
$$
To prove this, we build two continuous functions
\begin{gather}
\Phi_\varepsilon:\Omega^-\to M_\varepsilon
\cap I_\varepsilon^{m_\infty+\delta}, \label{e3.1} \\
\beta:M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}\to \Omega^+, \label{e3.2}
\end{gather}
where
$$
\Omega^-=\{x\in \Omega:d(x,\partial\Omega)<r\}, \quad
\Omega^+=\{x\in \mathbb{R}^3:d(x,\partial\Omega)<r\}
$$
with $r>0$ small enough so that
$\operatorname{cat}(\Omega^-)=\operatorname{cat}(\Omega^+)
=\operatorname{cat}(\Omega)$.
Following the idea in \cite{Be4}, we can find two functions
$\Phi_\varepsilon$ and $\beta$ such that
$\beta\circ\Phi_\varepsilon:\Omega^-\to \Omega^+$
is homotopic to the immersion $i:\Omega^-\to \Omega^+$.
By Remark \ref{rmk2.2} we obtain the inequality which completes
 the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Remark \ref{rmk2.4}, \eqref{e3.1} and \eqref{e3.2}, we have
$$
P_t(M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta})
=P_t(\Omega)+Z(t)
$$
where $Z(t)$ is a polynomial with nonnegative coefficients.
Since $\inf_\varepsilon m_\varepsilon=c>0$, we have
\begin{gather} \label{e3.3}
P_t(I_\varepsilon^{m_\infty+\delta},I_\varepsilon^{\frac{c}{2}})
=tP_t(M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}),
\\
P_t(H_\varepsilon,I_\varepsilon^{m_\infty+\delta})
=t(P_t(I_\varepsilon^{m_\infty+\delta},I_\varepsilon^{\frac{c}{2}})-t)\,.
\label{e3.4}
\end{gather}
By Morse theory we have
\begin{equation}
\sum_{u\in \mathcal{K}_\varepsilon}t^{\mu(u)}
=P_t(H_\varepsilon,I_\varepsilon^{m_\infty+\delta})
+P_t(I_\varepsilon^{m_\infty+\delta},I_\varepsilon^{\frac{c}{2}})
+(1+t)Q_\varepsilon(t)\label{e3.5}
\end{equation}
where $\mathcal{K}_\varepsilon$ be the set of critical points of
$I_\varepsilon$, $\mu(u)$ is the Morse index of $u$, $Q_\varepsilon(t)$
is a polynomial with nonnegative coefficients.
Using this relation with \eqref{e3.3}--\eqref{e3.5}, we obtain
\begin{equation}
\sum_{u\in \mathcal{K}_\varepsilon}t^{\mu(u)}
=tP_t(\Omega)+t^2(P_t(\Omega)-1)+t(1+t)Q_\varepsilon(t)\label{e3.6}
\end{equation}
Theorem \ref{thm1.2} easily follows by evaluating the power series \eqref{e3.6} for $t=1$.
\end{proof}


\section{The function $\Phi_\varepsilon$}

For  $\xi\in \Omega^-$ we define the function
\begin{equation}
\omega_{\xi,\varepsilon}(x)=U_\varepsilon(x-\xi)\chi_r(|x-\xi|)\label{e4.1}
\end{equation}
where $\chi_r$ is a smooth cut off function $\chi_r\equiv 1$ for
 $t\in[0,\frac{r}{2})$, $\chi_r\equiv 0$ for $t>r$ and
$|\chi_r'(t)|\leq 2/r$.

We  define $\Phi_\varepsilon:\Omega^-\to M_\varepsilon$ by
$$
\Phi_\varepsilon(\xi)=t_\varepsilon(\omega_{\xi,\varepsilon})
\omega_{\xi,\varepsilon}(x)
$$

\begin{remark} \label{rmk4.1} \rm
 We have that the following limits hold uniformly with respect to $\xi\in\Omega^-$,
$$
\|\omega_{\xi,\varepsilon}\|_\varepsilon\to \|U\|,\quad
|\omega_{\xi,\varepsilon}|_{\varepsilon,p}\to |U|_p\,.
$$
\end{remark}

\begin{proposition} \label{prop4.2}
For any $\varepsilon>0$ the map $\Phi_\varepsilon$ is continuous.
Moreover for any $\delta>0$
there exists $\varepsilon_0>0$ such that if $\varepsilon<\varepsilon_0$
then $I_\varepsilon(\Phi_\varepsilon(\xi))<m_\infty+\delta$.
\end{proposition}

\begin{proof}
Obviously, $\Phi_\varepsilon$ is continuous.
We claim that $t_\varepsilon(\omega_{\xi,\varepsilon})\to 1$
uniformly with respect to $\xi\in \Omega^-$.
In fact, by Lemma \ref{lem3.1} $t_\varepsilon(\omega_{\xi,\varepsilon})$
is the unique solution of
$$
at\|\omega_{\xi,\varepsilon}\|_\varepsilon^2
+t|\omega_{\xi,\varepsilon}|_{\varepsilon,2}^2+bt^3
\|\omega_{\xi,\varepsilon}\|_\varepsilon^4
=t^{p}|\omega_{\xi,\varepsilon}^+|_{\varepsilon,p+1}^{p+1}
$$
By Remark \ref{rmk4.1} we have the claim.

Now, by Remark \ref{rmk4.1} and the above claim we have
\begin{align*}
&I_\varepsilon(\Phi_\varepsilon(\xi))\\
&= (\frac{1}{2}-\frac{1}{p+1}) at_\varepsilon^2
 \|\omega_{\xi,\varepsilon}\|_\varepsilon^2+(\frac{1}{2}-\frac{1}{p+1})
t_\varepsilon^2|\omega_{\xi,\varepsilon}|_{\varepsilon,2}^2
 +(\frac{1}{4}-\frac{1}{p+1})bt_\varepsilon^4
 \|\omega_{\xi,\varepsilon}\|_\varepsilon^4\\
&\to (\frac{1}{2}-\frac{1}{p+1)})a\|U\|^2+(\frac{1}{2}
 -\frac{1}{p+1})|U|^2+(\frac{1}{4}-\frac{1}{p+1})b\|U\|^4
=m_\infty
\end{align*}
this completes the proof.
\end{proof}

\begin{remark} \label{rmk4.3}\rm
Note that $\limsup_{\varepsilon\to 0}m_\varepsilon\leq m_\infty$.
\end{remark}

\section{The map $\beta$}

For  $u\in M_\varepsilon$ we can define a point $\beta(u)\in \mathbb{R}^3$ by
$$
\beta(u)=\frac{\int_\Omega x|u^+|^{p+1}dx}{\int_\Omega|u^+|^{p+1}dx}
$$
The function $\beta$ is well defined in $M_\varepsilon$ since if
$u\in M_\varepsilon$ then $u^+\neq 0$.

We shall prove that if $u\in M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}$
then $\beta(u)\in \Omega^+$.
First of all we consider partitions of the compact manifold $\Omega$.
Given $\varepsilon>0$, a finite partition
$P_\varepsilon=\{P_j^\varepsilon\}_{j\in \Lambda_\varepsilon}$ is called a ``good''
partition if: for any $j\in \Lambda_\varepsilon$ the set
$P_j^\varepsilon$ is closed;
$P_i^\varepsilon\cap P_j^\varepsilon\subseteq\partial P_i^\varepsilon
\cap\partial P_j^\varepsilon$ for $i\neq j$; there exist
$r_1(\varepsilon), r_2(\varepsilon)>0$ such that, for any $j$, there exists
a point $q_j^\varepsilon\in P_j^\varepsilon$ such that
\[
B(q_j^\varepsilon,\varepsilon)\subset P_j^\varepsilon
\subset B(q_j^\varepsilon,r_2(\varepsilon))
\subset B(q_j^\varepsilon,r_1(\varepsilon)),
\]
 with $r_1(\varepsilon)\geq r_2(\varepsilon)\geq C\varepsilon$ for some
positive constant $C$; lastly, there exists a finite number $\iota\in N$
such that every $x\in \Omega$ is contained in at most $\iota$
balls $B(q_j^\varepsilon,r_1(\varepsilon))$, where $\iota$ does not
depends on $\varepsilon$.

\begin{lemma} \label{lem5.1}
There exists $\gamma>0$ such that, for any $\delta>0$ and any
$\varepsilon\in (0,\varepsilon_0(\delta))$ where
$\varepsilon_0(\delta)$ is as in Proposition \ref{prop4.2}, given any ``good''
partition $P_\varepsilon$ of the domain $\Omega$ and for any
$u\in M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}$ there exists a
set $P_j^\varepsilon$ such that
$$
\frac{1}{\varepsilon^3}\int_{P_j^\varepsilon}|u^+|^{p+1}dx\geq \gamma
$$
\end{lemma}

\begin{proof}
Taking into account that $G_\varepsilon(u)=0$ we have
\begin{align*}
a\|u\|_\varepsilon^2
&\leq |u^+|_{\varepsilon,p+1}^{p+1}
 =\sum_j\frac{1}{\varepsilon^3}\int_{P_j^\varepsilon}|u^+|^{p+1}dx\\
&\leq \sum_j|u_j^+|_{\varepsilon,p+1}^{p+1}
 =\sum_j|u_j^+|_{\varepsilon,p+1}^{p-1}|u_j^+|_{\varepsilon,p+1}^2\\
&\leq \max_j\{|u_j^+|_{\varepsilon,p+1}^{p-1}\}\sum_j|u_j^+|_{\varepsilon,p+1}^2
\end{align*}
where $u_j^+$ is the restriction of the function $u^+$ on the set $P_j^\varepsilon$.

Arguing as in \cite{Be1}, we prove that there exists a constant $C>0$ such that
$$
\sum_j|u_j^+|_{\varepsilon,p+1}^2\leq C\iota\|u^+\|_\varepsilon^2,
$$
thus 
$$
\max_j\{|u_j^+|_{\varepsilon,p+1}^{p-1}\}\geq\frac{a}{C\iota}
$$
that concludes the proof.
\end{proof}

\begin{proposition} \label{prop5.2} 
For any $\eta\in(0,1)$ there exists $\delta_0>0$ such that for any 
$\delta\in(0,\delta_0)$ and any $\varepsilon\in (0,\varepsilon_0(\delta))$ 
as in Proposition \ref{prop4.2}, for any 
$u\in M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}$ 
there exists a point $q=q(u)\in \Omega$ such that
$$
\frac{1}{\varepsilon^3}\int_{B(q,\frac{r}{2})}|u^+|^{p+1}dx
>(1-\eta)\frac{2(p+1)}{p-1}\Big(4m_\infty+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}\Big)
$$
\end{proposition}

\begin{proof} 
We prove only the proposition for any 
$u\in M_\varepsilon\cap I_\varepsilon^{m_\varepsilon+2\delta}$. 
Indeed, by this result and by Remark \ref{rmk4.3} we get
$$
\lim_{\varepsilon\to 0}m_\varepsilon=m_\infty
$$
Hence it holds 
$I_\varepsilon^{m_\infty+\delta}\subset I_\varepsilon^{m_\varepsilon+2\delta}$
 for $\delta,\varepsilon$ small enough. So the thesis holds.

We argue by contradiction. Suppose that there exists $\eta\in (0,1)$ 
such that we can find vanishing sequences $\{\delta_k\}$,
$\{\varepsilon_k\}$ and a sequence 
$\{u_k\}\subset M_{\varepsilon_k}\cap I_\varepsilon^{m_{\varepsilon_k}+2\delta_k}$ 
such that
\begin{equation}
\begin{aligned}
m_{\varepsilon_k}
&\leq I_{\varepsilon_k}(u_k)\\
&=(\frac{1}{2}-\frac{1}{p+1})a\|u_k\|_{\varepsilon_k}^2
+(\frac{1}{2}-\frac{1}{p+1})|u_k|_{\varepsilon_k,2}^2
 +(\frac{1}{4}-\frac{1}{p+1})b\|u_k\|_{\varepsilon_k}^4 \\
&\leq m_{\varepsilon_k}+2\delta_k
 \leq m_\infty+3\delta_k.
\end{aligned}\label{e5.1}
\end{equation}
for $k$ large enough, and for any $q\in \Omega$,
\begin{equation}
\frac{1}{\varepsilon_k^3}\int_{B(q,\frac{r}{2})}|u_k^+|^{p+1}dx
\leq(1-\eta)\frac{2(p+1)}{p-1}(4m_\infty
+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}). \label{e5.2}
\end{equation}

By Ekeland variational principle and by definition of $M_{\varepsilon_k}$ 
we can assume that
\begin{equation}
I'_{\varepsilon_k}(u_k)\to 0.\label{e5.3}
\end{equation}
By Lemma \ref{lem5.1}, there exists a set $P_k^{\varepsilon_k}\in P_{\varepsilon_k}$
such that
\begin{equation}
\frac{1}{\varepsilon^3}\int_{P_k^{\varepsilon_k}}|u_k^+|^{p+1}dx
\geq\gamma.\label{e5.4}
\end{equation}
So we can choose a point $q_k\in \overset{\circ}{{P_k^{\varepsilon_k}}}$,
and define, for $z\in \Omega_{\varepsilon_k}=\frac{1}{\varepsilon_k}(\Omega-q_k)$,
$$
\omega_k(z)=u_k(\varepsilon_kz+q_k)=u_k(x),
$$
where $x\in \Omega$. We obtain that $\omega_k\in H_0^1(\Omega_{\varepsilon_k})$.
By \eqref{e5.1}, we have
$$
\|\omega_k\|_{H_0^1(\Omega_{\varepsilon_k})}^2\leq C.
$$
So we obtain $\omega_k\rightharpoonup \omega$ in $H_{loc}^1(\mathbb{R}^3)$,
$\omega_k\to \omega$ in $L_{loc}^{s}(\mathbb{R}^3)(2\leq s<6)$ and
$\|\omega_k\|^2\to A_1$. Thus we prove that $\omega\not\equiv 0$
and $A_1>0$ by \eqref{e5.4}.

Next we claim 
\begin{equation}
\lim_{k\to\infty}\frac{dist(q_k,\partial\Omega)}{\varepsilon_k}
=\infty.\label{e5.5}
\end{equation}
We argue by contradiction. Suppose that
$$
\lim_{k\to\infty}\frac{dist(q_k,\partial\Omega)}{\varepsilon_k}=d<\infty.
$$
It is easy to verify that $\omega$ is a solution of
\begin{equation}
\begin{gathered}
 -(a+bA_1)\Delta u+u=|u|^{p-1}u,\quad x\in \mathbb{R}^3_+ \\
    u(x)=0, \quad  x\in \partial \mathbb{R}^3_+
    \end{gathered} \label{ePpi}
\end{equation}
where $\mathbb{R}^3_+$ is a half space. We know that \eqref{ePpi}
 has no nontrivial solution from the work by \cite{Ai,Es}.
So $\omega\equiv 0$, this contradicts with $\omega\not\equiv 0$.
This concludes the claim.

By \eqref{e5.5}, $\Omega_{\varepsilon_k}$ converges to the whole space 
$\mathbb{R}^3$ as $k\to \infty$. Using \eqref{e5.1} \eqref{e5.3} 
and computing, we have
\begin{equation}
I_\infty(\omega_k)\to m_\infty,\quad
I'_\infty(\omega_k)\to0.\label{e5.6}
\end{equation}
This implies $\{\omega_k\}\subset M_\infty$ is a minimizing sequences
for $m_\infty$. Arguing as in \cite{He2}, we have $\omega_k\to \omega$,
$\omega\in M_\infty$, $I_\infty(\omega)=m_\infty$ and $I'_\infty(\omega)=0$.

By using \eqref{e5.6} and  Pohozaev identity, we have
\begin{gather*}
\frac{a}{2}\|\omega_k\|^2+\frac{1}{2}|\omega_k|_2^2+\frac{b}{4}\|\omega_k\|^4
-\frac{1}{p+1}|\omega_k|_{p+1}^{p+1}=m_\infty+o_k(1),
\\
a\|\omega_k\|^2+|\omega_k|_2^2+b\|\omega_k\|^4-|\omega_k|_{p+1}^{p+1}=o_k(1)
\\
\frac{a}{2}\|\omega_k\|^2+\frac{3}{2}|\omega_k|_2^2
 +\frac{b}{2}\|\omega_k\|^4-\frac{3}{p+1}|\omega_k|_{p+1}^{p+1}=o_k(1)
\end{gather*}
Thus, we obtain that  
$$
|\omega_k|_{p+1}^{p+1}\to \frac{2(p+1)}{p-1}
\Big(4m_\infty+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}\Big)
$$
So by $\omega_k\to \omega$, for $T$ and $k$ large enough, we have
$$
\int_{B(0,T)}|\omega_k^+|^{p+1}dz
>(1-\eta)\frac{2(p+1)}{p-1}\Big(4m_\infty+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}\Big)
$$

On the other hand by \eqref{e5.2} and the definition of $\omega_k$, 
for any $T>0$ we have, for $k$ large enough,
\begin{align*}
\int_{B(0,T)}|\omega_k^+|^{p+1}dz
&\leq\frac{1}{\varepsilon_k^3}\int_{B(q_k,\varepsilon_kT)}|u_k|^{p+1}dx\\
&\leq \frac{1}{\varepsilon_k^3}\int_{B(q_k,\frac{r}{2})}|u_k|^{p+1}dx\\
&\leq (1-\eta)\frac{2(p+1)}{p-1}(4m_\infty+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}).
\end{align*}
This leads to a contradiction.
\end{proof}

\begin{proposition} \label{prop5.3} 
There exists $\delta_0>0$ such that for any $\delta\in (0,\delta_0)$ 
and any $\varepsilon\in (0,\varepsilon(\delta_0))$ as in Proposition \ref{prop5.2}, 
for any $u\in M_\varepsilon\cap I_\varepsilon^{m_\infty+\delta}$ it holds
 $\beta(u)\in \Omega^+$. Moreover the composition 
$$
\beta\circ\Phi_\varepsilon:\Omega^-\to \Omega^+
$$
is homotopic to the immersion $i:\Omega^-\to \Omega^+$.
\end{proposition}

\begin{proof}
Arguing by contradiction, we suppose that there exist sequences 
$\{\delta_k\},\{\varepsilon_k\}\subset R$ and 
$\{u_k\}\subset M_{\varepsilon_k}\cap I_\varepsilon^{m_\infty+\delta_k}$ 
such that $\delta_k,\varepsilon_k \to 0^+$, as 
$k\to \infty$, and $\beta(u_k)\not\in \Omega^+$ for all $k$.

By Ekeland variational principle and by definition of $M_{\varepsilon_k}$ 
we can assume that $I'_{\varepsilon_k}(u_k)\to 0$. 
So by Proposition \ref{prop5.2} we can find $q_k\in \Omega$ such that
$$
\frac{\frac{1}{\varepsilon_k^3}\int_{B(q_k,\frac{r}{2})}|u_k|^{p+1}dx}{\frac{1}{\varepsilon_k^3}\int_\Omega|u_k|^{p+1}dx}
>\frac{(1-\eta)\frac{2(p+1)}{p-1}
\big(4m_\infty+\frac{2a^2-2a\sqrt{a^2+3bm_\infty}}{b}\big)}
{\frac{2(p+1)}{p-1}\big(4(m_\infty+\delta_k)
+\frac{2a^2-2a\sqrt{a^2+3b(m_\infty+\delta_k)}}{b}\big)}
$$
Finally,
\begin{align*}
&|\beta(u_k)-q_k|\\
&\leq  \frac{|\frac{1}{\varepsilon_k^3}
 \int_{\Omega}(x-q_k)|u_k|^{p+1}dx|}{\frac{1}{\varepsilon_k^3}
 \int_\Omega|u_k|^{p+1}dx}\\
&\leq \frac{|\frac{1}{\varepsilon_k^3}\int_{B(q_k,\frac{r}{2})}
 (x-q_k)|u_k|^{p+1}dx|}{\frac{1}{\varepsilon_k^3}\int_\Omega|u_k|^{p+1}dx}
 +\frac{|\frac{1}{\varepsilon_k^3}\int_{\Omega
 \backslash B(q_k,\frac{r}{2})}(x-q_k)|u_k|^{p+1}dx|}
 {\frac{1}{\varepsilon_k^3}\int_\Omega|u_k|^{p+1}dx}\\
&\leq \frac{r}{2}+2\operatorname{diam}(\Omega)
\Big(1-\frac{(1-\eta)\frac{2(p+1)}{p-1}(4m_\infty+\frac{2a^2
-2a\sqrt{a^2+3bm_\infty}}{b})}{\frac{2(p+1)}{p-1}
(4(m_\infty+\delta_k)+\frac{2a^2-2a\sqrt{a^2
+3b(m_\infty+\delta_k)}}{b})}\Big)
\end{align*}
The above expression implies that $\beta(u_k)\in \Omega^+$,
 which contradicts $\beta(u_k)\not\in \Omega^+$.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the Shanxi University Technology research 
and development (project 2013158).


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\end{document}