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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 98, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/98\hfil Sublinear Kirchhoff equations]
{Infinitely many solutions for sublinear Kirchhoff
equations in $\mathbb{R}^N$ with sign-changing potentials}

\author[A. Bahrouni \hfil EJDE-2013/98\hfilneg]
{Anouar Bahrouni}  % in alphabetical order

\address{Anouar Bahrouni \newline
Mathematics Department, University of Monastir,
Faculty of Sciences, 5019 Monastir, Tunisia}
\email{bahrounianouar@yahoo.fr, Fax + 216 73 500 278}

\thanks{Submitted March 6, 2013, Published April 16, 2013.}
\subjclass[2000]{35J60, 35J91, 58E30}
\keywords{Kirchhoff equations; symmetric Mountain Pass Theorem;
\hfill\break\indent  infinitely many solutions}

\begin{abstract}
 In this article we study the  Kirchhoff equation
 $$
  -\Big(a+b \int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u
  = K(x)|u|^{q-1}u, \quad\text{in }\mathbb{R}^N,
 $$
 where $N\geq 3$, $0<q<1$, $a,b>0$ are constants and $K(x), V(x)$ both
 change sign in $\mathbb{R}^N$. Under appropriate assumptions on
 $V(x)$ and $K(x)$, the existence of infinitely many solutions is
 proved by using the symmetric Mountain Pass Theorem.
\end{abstract}

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

In this article, we study the existence of infinitely many solutions 
of the  nonlinear Kirchhoff equation
\begin{equation}\label{3main}
\begin{gathered}
-\Big(a+b \int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u
= K(x)|u|^{q-1}u, \quad\text{in }\mathbb{R}^N,\\
u\in H^1(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N),
\end{gathered}
\end{equation}
where $N\geq 3$, $0<q<1$,  $a,b$ are positive constants and 
$K(x), V(x)\in L^{\infty}(\mathbb{R}^N)$ both change sign in 
$\mathbb{R}^N$ and satisfies some  conditions specified below.

If in \eqref{3main}, we set $V(x)=0$ and replace $\mathbb{R}^N$ by a bounded domain
 $\Omega \subset \mathbb{R}^N$, then \eqref{3main} reduces to the 
 problem
\begin{equation}\label{3mainb}
\begin{gathered}
-\Big(a+b \int_{\Omega}|\nabla u|^2dx\Big)\Delta u=f(x,u) \quad\text{in }\Omega,
\\
u=0 \quad\text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $f(x,u)\in C(\mathbb{R}^N\times \mathbb{R}, \mathbb{R})$.
Problem \eqref{3mainb} is related to the stationary analogue of the 
Kirchhoff equation
$$
u_{tt}-\Big(a+b \int_{\Omega}|\nabla_{x} u|^2dx\Big)\Delta_{x} u
=f(x,u)
$$
which was proposed by Kirchhoff in 1883 \cite{k2} as a generalization 
of the well-known d'Alembert's wave equation
$$
\rho \frac{\partial ^2u}{\partial^2t}
-\Big(\frac{P_0}{h}+\frac{E}{2L} \int_0^{L}
|\frac{\partial u}{\partial x}|^2dx\Big)
\frac{\partial ^2u}{\partial^2x}=f(x,u)
$$
for free vibrations of elastic strings.
 Kirchhoff's model takes into account the changes in length of the 
string produced by transverse vibrations. Some early classical investigations 
of Kirchhoff equations can be seen in Bernstein \cite{b2} and 
Poho\^zaev \cite{p1}. Problem \eqref{3main} called the attention of several 
researchers mainly after the work of Lions \cite{l1}, where a functional 
analysis approach was proposed. Recently, problems like type \eqref{3main} 
have been investigated by several authors;
see \cite{a1,a2,c2,h1,h2,h3,h4,m1,n1,w1}.


 Ma and Rivera \cite{m1} obtained a positive solutions of \eqref{3mainb} 
by using variational methods.  He and Zou \cite{h3} showed that 
problem \eqref{3mainb} admits infinitely many solutions by using the 
local minimum methods and the Fountain Theorem.
Very recently, some authors have studied the Kirchhoff equation on the 
whole space $\mathbb{R}^N$. Since \eqref{3main} is set on $\mathbb{R}^N$, 
it is well known that the Sobolev embedding 
$H^1(\mathbb{R}^N)\hookrightarrow L^{m}(\mathbb{R}^N)$
$(2\leq m\leq 2^{\ast}=\frac{2N}{N-2})$ is not compact and then it is usually 
difficult to prove that a minimizing sequence or a Palais-Smale sequence 
is strongly convergent if we seek solution of \eqref{3main} by variational 
methods. If $V(x)$ is radial, as in \cite{n1,w1} we can avoid the lack 
of compactness of Sobolev embedding by looking for solution of \eqref{3main} 
in the subspace of radial functions of $D^{1,2}(\mathbb{R}^N)$ since the 
embedding is compact.  Nie and Wu \cite{n1}  studied a 
Schr\"odinger-Kirchhoff-type equation with radial potential and they  proved 
the existence of infinitely many solutions by using a symmetric Mountain 
Pass Theorem. On the other hand, Alves and Figueiredo \cite{a1}  studied a
 periodic Kirchhoff equation in $\mathbb{R}^N$. They proved the existence 
of a nontrivial solution when the nonlinearity is in subcritical and 
critical case.  Liu  and He \cite{h4} proved the existence of infinitely
 many high-energy solutions where $V(x)\geq 0$ and the nonlinearity 
is superlinear.
We remark that when $a=1$ and $b=0$, problem \eqref{3main} can be rewritten 
as the  well-known Schr\"odinger equation
\begin{equation}\label{sch}
-\Delta u+ V(x)u=K(x)|u|^{q-1}u, \quad x\in \mathbb{R}^N.
\end{equation}
For this equation, there is a large body of literature on the existence 
and multiplicity of solutions; we  for example \cite{b1,c1,d1,k1,w2}
and the references therein.
Motivated by the above fact, in this paper our aim is to study the existence 
of infinitely many nontrivial solutions for \eqref{3main} when $0<q<1$ 
and $K(x), V(x)$ both change sign. Our tool is the symmetric Mountain 
Pass Theorem.

To state our main result we require the following assumptions:
\begin{itemize}
\item[(A1)] $V\in L^{\infty}(\mathbb{R}^N)$ and there exist
 $\beta,R_0>0$ such that
$V(x)\geq \beta$ for all $|x|\geq R_0$;

\item[(A2)] $K\in L^{\infty}(\mathbb{R}^N)$ and there exist 
$\alpha, R_1, R_2>0$, $y_0=(y_1,\cdots, y_{N})\in \mathbb{R}^N$ such that
$K(x)\leq -\alpha$ for all $|x| \geq R_1$ and $K(x)>0$ for all 
$x\in B(y_0, R_2)$.
\end{itemize}
Our main result reads as follows.

\begin{theorem}\label{thm1.1}
Under assumptions {\rm (A1), (A2)}, problem \eqref{3main} admits 
infinitely many nontrivial solutions.
\end{theorem}

In the next section we give some notation and preliminary results;
and in section 3, we prove theorem \eqref{thm1.1}.

\section{Preliminaries}

We will us the following notation: Let
 \[
 \|u\|_m=\Big( \int_{\mathbb{R}^N}|u(x)|^{m} dx\Big)^{1/m}, \quad
 1\leq m < +\infty.
\]
Let $2^{\ast}=\frac{2N}{N-2}$ for all $N\geq 3$.
Let  $B_R$ denote the ball centred in zero of radius $R>0$ in $\mathbb{R}^N$ 
and $B_R^{c}=\mathbb{R}^N\backslash B_R$. 
Let  $F'(u):$ the Fr\'echet derivative of $F$ at $u$.
For  $s$, be the Sobolev constant in
 $$ 
\|u\|_{2^{\ast}}\leq s \|\nabla u\|_2, \quad \forall u\in H^1(\mathbb{R}^N).
$$
 Let $E=H^1(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N)$, $0<q<1$, endowed  
with the norm
 $$
\|u\|=\|\nabla u\|_2+\|u\|_{q+1},
$$
The space $E$ becomes a reflexive Banach space.

Problem \eqref{3main} has a variational structure. Indeed we consider the 
functional $I: E\to \mathbb{R}$ defined by
$$
I(u)=\frac{b}{4}\Big( \int_{\mathbb{R}^N}|\nabla u|^2 dx\Big)^2
+ \frac{a}{2} \int_{\mathbb{R}^N}|\nabla u|^2 dx
+\frac{1}{2} \int_{\mathbb{R}^N}V(x) u^2 dx
-\frac{1}{q+1}\int_{\mathbb{R}^N} K(x) |u |^{q+1} dx.
$$
As is well known, $I$ is of class $C^1$ on $E$ and any critical point of $I$ 
is a solution of \eqref{3main}.

A functional  $I$  is said to satisfy the Palais-Smale condition (PS, for short)
if for very sequence $(u_n)$ such that
$$
I(u_n) \text{ is bounded, and }\|I'(u_n)\|\to 0,
$$
there is a convergent subsequence of $(u_n)$.

Before proving Theorem \eqref{thm1.1}, we give the symmetric 
Mountain Pass Theorem.

\begin{definition} \label{def2.1} \rm
 Let $E$ be a Banach space and $A$ a subset of $E$. Set $A$ is said to be 
symmetric if $u\in E$ implies $-u\in E$. For a closed symmetric set 
$A$ which does not contain the origin, we define a genus $\gamma (A)$ of $A$ 
by the smallest integer $k$ such that there exists an odd continuous mapping 
from $A$ to $\mathbb{R}^{k} \backslash  \{0\}$. 
If there does not exist such a $k$, we define $\gamma (A)=\infty$. 
We set $\gamma ( \emptyset )=0$. Let $\Gamma_k$ denote the family
of closed symmetric subsets $A$ of $E$ such that $0\notin A$ and 
$\gamma(A)\geq k$.
 \end{definition}

Now we give the symmetric Mountain Pass Theorem \cite{a2} which improved by 
Kajikiya \cite{k1} to obtain the following Theorem.

\begin{theorem} \label{thm2.2}
 Let $E$ be an infinite dimensional Banach space and $I\in C^1(E, \mathbb{R})$ 
satisfy:
\begin{itemize}
\item[(1)] $I$ is even, bounded from below, $I(0)=0$ and $I$ satisfies 
 the Palais-Smale condition.
\item[(2)] For each $k\in \mathbb{N}$, there exists an $A_k\in \Gamma_k$
 such that
$$  
\sup_{u\in A_k }  I(u)< 0.
$$
\end{itemize}
Then either of the following tow conditions holds:
\begin{itemize}
\item[(i)] There exists a sequence $(u_k)$ such that 
$I'(u_k)=0$, $I(u_k)<0$ and $(u_k)$ converges to zero; or

\item[(ii)] There exist two sequences $(u_k)$ and $(v_k)$ such that 
$I'(u_k)=0$, $I(u_k)=0$, $u_k\neq 0$, $\lim_{k\to +\infty}u_k=0$, 
$I'(v_k)=0$, $I(v_k)<0$, $\lim_{k\to +\infty} I(v_k)=0$ and 
$(v_k)$ converges to a non-zero limit.
\end{itemize}
\end{theorem}

\section{Proof of Theorem \eqref{thm1.1}}

\begin{lemma}\label{kbelow}
Under assumptions {\rm (A1), (A2)}, the functional $I$ is bounded from below.
\end{lemma}

\begin{proof}
By (A1),  (A2) and H\"older inequality, we have
\begin{align*}
I(u)
&=\frac{b}{4}\Big( \int_{\mathbb{R}^N}|\nabla u|^2 dx\Big)^2
 + \frac{a}{2} \int_{\mathbb{R}^N}|\nabla u|^2 dx
 +\frac{1}{2} \int_{\mathbb{R}^N}V(x) u^2 dx
 -\frac{1}{q+1}\int_{\mathbb{R}^N} K(x) |u |^{q+1} dx \\
&\geq \frac{b}{4} ( \int_{\mathbb{R}^N} |\nabla u|^2dx)^2
 -\frac{1}{2} \int_{\mathbb{R}^N}V^{-}(x)u^2 dx
 -\frac{1}{q+1} \int_{\mathbb{R}^N} K^{+}(x) | u |^{q+1} dx\\
&\geq \frac{b}{4} \|\nabla u\|_2^{4}-\frac{s^2}{2}\| V^{-}\|_{N/2}
 \| \nabla u\|_2^2- s^{q+1} \| K^{+}\|_{\frac{2^{\ast}}{2^{\ast}-q-1}}
\| \nabla u\|_2^{q+1}.
\end{align*}
Since $0<q<1$, we conclude the proof.
\end{proof}

\begin{lemma}\label{kbounded}
Assume {\rm (A1), (A2)}  hold. Then, any $(PS)$ sequence 
$(u_n)$ of $I$ is bounded in $E$.
\end{lemma}

\begin{proof}
Let $(u_n)$ be a (PS) sequence of $I$. Then, there exists a positive 
constant $c>0$ such that
\begin{align*}
c 
&\geq I(u_n)\\
&=\frac{b}{4}\Big( \int_{\mathbb{R}^N}|\nabla u_n|^2dx\Big)^2
 +\frac{a}{2}  \int_{\mathbb{R}^N} |\nabla u_n|^2dx
 + \frac{1}{2} \int_{\mathbb{R}^N}V(x) u_n^2 dx \\
&\quad -\frac{1}{q+1} \int_{\mathbb{R}^N}K(x)|u_n|^{q+1} dx \\
&\geq\frac{b}{4} ( \int_{\mathbb{R}^N} |\nabla u_n|^2dx)^2
 -\frac{1}{2} \int_{\mathbb{R}^N}V^{-}(x)u_n^2 dx
 -\frac{1}{q+1} \int_{\mathbb{R}^N} K^{+}(x) | u_n |^{q+1} dx\\
&\geq \frac{b}{4} \|\nabla u_n\|_2^{4}-\frac{s^2}{2}\| V^{-}\|_{N/2}
 \| \nabla u_n\|_2^2- s^{q+1}\| K^{+}\|_{\frac{2^{\ast}}{2^{\ast}-q-1}}
 \| \nabla u_n\|_2^{q+1}.
\end{align*}
 Hence, there exists $\gamma_0>0$ such that
\begin{equation}\label{kgrad}
\|\nabla u_n\|_2\leq \gamma_0, \quad \forall  n \in \mathbb{N}.
\end{equation}
On the other hand, there exists $c>0$ such that
\begin{align*}
c+\frac{\|u_n\|}{4} 
&\geq -\frac{1}{4} \langle I'(u_n),u_n\rangle+I( u_n)\\
&=\frac{a}{4} \int_{\mathbb{R}^N}|\nabla u_n|^2dx
 +\frac{1}{4} \int_{\mathbb{R}^N}V(x)u_n^2dx
 +(\frac{1}{4}-\frac{1}{q+1})  \int_{\mathbb{R}^N}K(x)|u_n|^{q+1}dx \\
& \geq (\frac{1}{q+1}-\frac{1}{4}) \int_{\mathbb{R}^N}(K^{-}(x)
 +\chi_{B_{R_1}}(x)) |u_n|^{q+1} dx \\
&\quad -(\frac{1}{q+1}-\frac{1}{4}) \int_{\mathbb{R}^N}(K^{+}(x)
 +\chi_{B_{R_1}}(x)) |u_n|^{q+1} dx
 -\frac{1}{4} \int_{\mathbb{R}^N}V^{-}(x)u_n^2dx\\
&\geq (\frac{1}{q+1}-\frac{1}{4}) \min(\alpha, 1)  
 \int_{\mathbb{R}^N}|u_n|^{q+1}(x) dx-\frac{s^2
 \|V^{-}\|_{N/2}}{4}\|\nabla u_n\|^2\\
&\quad -s^{q+1}(\frac{1}{q+1}-\frac{1}{4})\|K^{+}
 +\chi_{B_{R_1}}\|_{\frac{2^{\ast}}{2^{\ast}-q-1}} \|\nabla u_n\|_2^{q+1}.
\end{align*}
Therefore, by using \eqref{kgrad}, we obtain
\begin{equation}\label{kq}
\|u_n\|_{q+1}\leq \gamma_1, \quad\text{for some } \gamma_1>0.
\end{equation}
Combining \eqref{kgrad} and \eqref{kq}, we conclude the proof.
\end{proof}

We need the following Lemma to prove that the Palais-Smale condition 
is satisfied for $I$ on $E$.

\begin{lemma}\label{apprendix}
Let $x$ and $y$ two arbitrary real numbers, then there exists a constant 
$c>0$ such that
\begin{equation}\label{xy}
\big||x+y|^{q+1}-|x|^{q+1}-|y|^{q+1}\big|\leq c |x|^qy
\end{equation}
\end{lemma}

\begin{proof}
If $x=0$, the inequality \eqref{xy} is trivial.
Suppose that $x\neq 0$. We consider the continuous function $f$ defined 
on $\mathbb{R}\backslash \{0\}$ by
$$
f(t)=\frac{|1+t|^{q+1}-|t|^{q+1}-1}{|t|}.
$$
Note that $ \lim_{|t|\to +\infty}f(t)=0$ and 
$\lim_{t\to  0\pm}f(t)=\pm (q+1)$. Then there exists a constant $c>0$ 
such that $|f(t)|\leq c$, for all $t\in \mathbb{R}\backslash \left\{0\right\}$. 
In particular $|f(\frac{y}{x})|\leq c$, so
$$
\big||1+\frac{y}{x}|^{q+1}-|\frac{y}{x}|^{q+1}-1\big|\leq c |\frac{y}{x}|,
$$
multiplying by $|x|^{q+1}$, we obtain the desired result.
\end{proof}

\begin{lemma}\label{kPalais-Smalee}
Assume that {\rm (a1), (A2)} hold. Then $I$ satisfies the Palais-Smale condition 
in $E$.
\end{lemma}

\begin{proof}
Let $(u_n)$ be a $(PS)$ sequence. By Lemma \eqref{kbounded},
 $(u_n)$ is bounded in $E$. Then there exists a subsequence 
$u_n\rightharpoonup u$ in $E$, $u_n\to u$ in $L^{p}_{Loc}(\mathbb{R}^N)$ 
for all  $1\leq p\leq 2^{\ast}$ and $u_n\to u$ a.e in $\mathbb{R}^N$.

By \cite{d1}, it is sufficient to prove that for any $\epsilon>0$, 
there exist $R_{3}>0$ and $n_0\in \mathbb{N}^{\ast}$ such that
$$ 
\int_{|x|\geq R_{3}}(|\nabla u_n|^2+|u_n|^{q+1})dx \leq \epsilon, \quad
\text{for all } R\geq R_{3} \text{ and } n\geq n_0.
$$
Let $\phi_R$ be a cut-off function so that $\phi_R=0$ on $B_{ \frac{R}{2}}$, 
$\phi_R=1$ on $B^{c}_R$, $0<\phi_R<1$ and
\begin{equation}\label{22}
|\nabla \phi_R|(x)\leq \frac{c}{R}, \quad\text{for all }x\in \mathbb{R}^N.
\end{equation}
We can easily remark that for any $u\in E$ and $R\geq 1$,
\begin{equation}\label{23}
\|\phi_R u\|\leq c \|u\|.
\end{equation}
Since $I'(u_n)\to 0$ in $E'$ as $n\to +\infty$, we know that for any 
$\epsilon>0$, there exists $n_0>0$ such that
$$
|\langle I'(u_n), \phi_Ru_n\rangle|\leq 
c \|I'(u_n)\|_{E'} \|u_n\|\leq \frac{\epsilon}{3}, \quad \forall n\geq n_0;
$$
that is, $n\geq n_0$. Then
\begin{align*}
&(a+b \int_{\mathbb{R}^N}|\nabla u_n|^2dx)
 \int_{\mathbb{R}^N}|\nabla u_n|^2\phi_R(x)dx\\
&+ \int_{\mathbb{R}^N}V(x)| u_n|^2\phi_R(x)dx
- \int_{\mathbb{R}^N}K(x)| u_n|^{q+1} \phi_R(x)dx\leq \frac{\epsilon}{3}.
\end{align*}
Hence,
\begin{equation} \label{k}
\begin{aligned}
&\int_{\mathbb{R}^N}(a|\nabla u_n|^2+(K^{-}
 +\chi_{B_{R_1}})(x)|u_n|^{q+1})\phi_R(x)dx \\
&\leq  \int_{\mathbb{R}^N} V^{-}(x)u_n^2\phi_Rdx
-a\int_{\mathbb{R}^N}u_n \nabla u_n \nabla \phi_Rdx\\
&\quad +\int_{\mathbb{R}^N}(K^{+}+\chi_{B_{R_1}})(x) |u_n|^{q+1}\phi_R dx
 +\frac{\epsilon}{3}.
\end{aligned}
\end{equation}
By H\"older inequality and \eqref{22}, there exists $R_4>0$ such that
\begin{equation}\label{25}
\int_{\mathbb{R}^N} u_n\nabla u_n\nabla \phi_Rdx \leq \frac{c}{R}
< \frac{\epsilon}{3}, \quad \forall |x|\geq R_4.
\end{equation}
 From (A1) and (A2), there exists $R_5>0$ such that
\begin{equation} \label{v}
\begin{aligned}
&\int_{\mathbb{R}^N} V^{-}(x)u_n^2 \phi_R dx
+\int_{\mathbb{R}^N}(K^{+}+\chi_{B_{R_1}})(x)|u_n|^{q+1} dx\\
&\leq c \|V^{-} \phi_R\|_{N/2}
 +c\|(K^{+}+\chi_{B_{ R_1}}) \phi_R\|_{\frac{2^{\ast}}{2^{\ast}-q-1}}\\
&\leq \frac{\epsilon}{3} \quad
 \text{for }  |x|\geq R_5.
\end{aligned}
\end{equation}
Put $R_3=\max(R_4,R_5)$. By \eqref{k}, \eqref{25} and \eqref{v}, we have
$$
\min(a,\min(\alpha,1))\int_{\mathbb{R}^N}(|\nabla u_n|^2+|u_n|^{q+1})\phi_R dx
\leq \epsilon.
$$
The proof is complete.
\end{proof}

\begin{lemma}\label{cubee}
Assume {\rm (A1), (A2)} hold. Then for each $k\in \mathbb{N}$, 
there exists an $A_k \in \Gamma_k$ such that
$$ 
\sup_{u\in A_k} I(u)<0.
$$
\end{lemma}

\begin{proof}
We use the following geometric construction introduced by 
 Kajikiya \cite{k1}:
Let $R_2$ and $y_0$ be fixed by assumption (A1) and consider the cube
$$
D(R_2)=\{(x_1,\cdots,x_{N})\in{\mathbb{R}^N}: |x_i-y_i|<R_2, \;
 1\leq i\leq N\}.
$$
Fix $k\in \mathbb{N}$ arbitrarily. Let $n \in\mathbb{N}$ be the
 smallest integer such that $n^N\geq k$. We divide $D(R_2)$ equally 
into $n^N$ small cubes, denote them by $D_i$ with $1\leq i \leq n^N$,
by planes parallel to each face of $D(R_2)$. The edge of $D_i$ has
 the length of $z=\frac{R_2}{n}$. We construct a new cubes 
$E_i$ in $D_i$ such that $E_i$ has the same center as that of $D_i$.
The faces of $E_i$ and $D_i$ are parallel and the edge of $E_i$
has the length of $\frac{z}{2}$. Then, we make a function 
$\psi_i$, $1\leq i \leq k$, such that
\begin{gather*}
\operatorname{supp} (\psi_i) \subset D_i, \quad
\operatorname{supp} (\psi_i)\cap \operatorname{supp}(\psi_{j})
=\emptyset \quad (i\neq j),\\
\psi_i(x)=1 \quad\text{for } x\in E_i, \quad 0\leq \psi_i(x)\leq 1,
\quad \forall x\in \mathbb{R}^N.
\end{gather*}
We denote
\begin{gather} \label{l'en V}
S^{k-1}=\{(t_1,\cdots, t_k)\in \mathbb{R}^{k}:  \max_{1\leq i \leq k}|t_i|=1\},\\
\label{l'en w}
W_k=\{\sum_{i=1}^{k}t_i\psi_i(x): (t_1,\cdots, t_k)\in S^{k-1}\}\subset E.
\end{gather}
Since the mapping $(t_1,\dots, t_k)\to \sum_{i=1}^{k} t_i \psi_i$
from $S^{k-1}$ to $W_k$ is odd and homeomorphic, then 
$\gamma(W_k)=\gamma (S^{k-1})=k$. $W_k$ is compact in $E$, then there is 
a constant $\alpha_k>0$ such that
$$
\|u\|^2 \leq \alpha_k \quad\text{for  all } u\in W_k.
$$
we need to recall the  inequality
\begin{equation}\label{interpolation}
\| u\|_2\leq c \|\nabla u\|_2^{r} \|u\|_{q+1}^{1-r}\leq c \|u\|
\end{equation}
with $r=\frac{2^{\ast} (q-1)}{2(2^{\ast}-q-1)}$. 
Then, there is a constant $c_k>0$ such that
$$
\|u \|_2^2 \leq c_k \quad\text{for  all } u\in W_k.
$$
Let $z>0$ and $u=\sum_{i=1}^{k}t_i \psi_i(x) \in W_k$,
\begin{equation} \label{pre maj}
I (z u) \leq z^{4}b\frac{\alpha_k^2}{4}+\frac{az^2}{2}\alpha_k+z^2
\frac{\|V\|_{\infty}}{2} c_k-\frac{1}{q+1}\sum_{i=1}^{k}
 \int_{D_i} K(x) |z t_i \psi_i|^{q+1}  dx.
\end{equation}
By \eqref{l'en V}, there exists  $j \in [1,k]$ such that $|t_{j}|=1$
 and $|t_i|\leq 1$ for $i\neq j$. Then
\begin{equation} \label{deux maj}
\begin{aligned}
&\sum_{i=1}^{k} \int_{D_i} K(x) |z t_i \psi_i|^{q+1}  dx \\
&= \int_{E_{j}}K(x) |z t_{j} \psi_{j}|^{q+1}  dx
+  \int_{D_{j}\backslash E_{j}}K(x) |z t_{j} \psi_{j}|^{q+1} dx
+\sum_{i\neq j}\int_{D_i}K(x) |z t_i \psi_i|^{q+1} dx\,.
\end{aligned}
\end{equation}
Since $\psi_{j}(x)=1$ for $x\in E_{j}$ and $|t_{j}|=1$, we have
\begin{equation} \label{troi maj}
 \int_{E_{j}}K(x) |z t_{j} \psi_{j}|^{q+1}  dx
= |z|^{q+1}  \int_{E_{j}}K(x)dx.
\end{equation}
On the other hand by  (A1) we obtain
\begin{equation}\label{quat maj}
 \int_{D_{j}\backslash E_{j}}K(x) |z t_{j} \psi_{j}|^{q+1}  dx
+  \sum_{i\neq j}  \int_{D_i}K(x) |z t_i \psi_i|^{q+1}  dx\geq 0.
\end{equation}
From \eqref{pre maj}, \eqref{deux maj}, \eqref{troi maj} and \eqref{quat maj},
we obtain
\begin{equation*}
\frac{I(z u)}{z^2}\leq z^2\frac{b\alpha_k^2}{4}+\frac{a\alpha_k}{2}
+ \frac{\|V\|_{\infty}}{2} c_k - \frac{|z|^{q+1}}{z^2}
\inf_{1\leq i \leq k}\Big( \int_{E_i}K(x) dx\Big).
\end{equation*}
It follows that
 $$
\lim_{z\to 0} \sup_{u\in W_k} \frac{I(z u)}{z^2}=-\infty.
$$
We fix $z$  small such that
$$
\sup\{I(u), u\in A_k\} < 0, \quad\text{where }A_k=z W_k \in  \Gamma_k.
$$
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \eqref{thm1.1}]
 Evidently, $I(0)=0$ and $I$ is an odd functional. Then by Lemmas
 \eqref{kbelow}, \eqref{kPalais-Smalee} and \eqref{cubee}, conditions 
(1) and (2) of Theorem \ref{thm2.2} are satisfied. Then,
by Theorem \ref{thm2.2}, problem \eqref{3main} admits an infinitely many
solutions $(u_k)\in E$ which converging to $0$ and $u_0$ can be supposed 
nonnegative since
$$
I(u_0)=I(|u_0|).
$$
\end{proof}


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