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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 10, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/10\hfil Sign-changing potential]
{Multiple solutions for  quasilinear elliptic equations
 with sign-changing potential}

\author[R. Wang, K. Wang, K. Teng \hfil EJDE-2016/10\hfilneg]
{Ruimeng Wang, Kun Wang, Kaimin Teng}

\address{Ruimeng Wang \newline
Department of Mathematics, Taiyuan
University of Technology, Taiyuan, \newline Shanxi 030024, China}
\email{wangruimeng112779@163.com}

\address{Kun Wang \newline
Department of Mathematics, Taiyuan
University of Technology, Taiyuan, \newline Shanxi 030024,  China}
\email{windwk0608@163.com}

\address{Kaimin Teng (Corresponding Author)\newline
Department of Mathematics, Taiyuan
University of Technology, Taiyuan, \newline Shanxi 030024,  China}
\email{tengkaimin2013@163.com}

\thanks{Submitted July 8, 2015. Published January 6, 2016.}
\subjclass[2010]{35B38, 35D05, 35J20}
\keywords{Quasilinear Schr\"odinger equation; symmetric mountain pass theorem;
\hfill\break\indent Cerami condition}

\begin{abstract}
 In this article, we study the quasilinear elliptic equation
 \[
 -\Delta_{p} u-(\Delta_{p}u^{2})u+ V (x)|u|^{p-2}u=g(x,u), 
 \quad x\in \mathbb{R}^N,
 \]
 where the potential $V(x)$ and the nonlinearity $g(x,u)$ are allowed to
 be sign-changing. Under some suitable assumptions on $V$ and $g$, we obtain
 the multiplicity of solutions by using minimax methods.
\end{abstract}

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

\section{Introduction}\label{00}

In this article, we are concerned with the multiplicity of nontrivial
solutions for the quasilinear elliptic equation
\begin{equation}\label{equ1-1}
-\Delta_{p} u-(\Delta_{p}u^{2})u+ V(x)|u|^{p-2}u=g(x,u),\quad  x\in \mathbb{R}^N,
\end{equation}
where $\Delta_{p} u:={\rm div}(|\nabla u|^{p-2} \nabla u)$ is the
 $p$-Laplacian operator with $2\leq p<N$, $N\geq3$, $V \in C(\mathbb{R}^N)$
and $g \in C(\mathbb{R}^N \times \mathbb{R})$ satisfy superlinear growth at infinity.

In recent years, there has been increasingly interest in the study of the
 quasilinear Schr\"{o}dinger equation
\begin{equation}\label{equ1-2}
-\Delta u- \Delta (u^{2})u + V(x)u= g(x,u),\quad x\in\mathbb{R}^N.
\end{equation}
Such equations are related to the existence of solitary wave solutions
for quasilinear Schr\"{o}dinger equations
\begin{equation}\label{equ1-3}
i\partial_{t} \psi= -\Delta \psi+ W(x)\psi- g(x, |\psi|^{2})\psi
-\kappa\Delta[\rho (|\psi|^{2})]\rho'(|\psi|^{2})\psi,
\end{equation}
where $\psi: \mathbb{R}\times \mathbb{R}^N \to \mathbb{C}$, $W(x)$ is a
given potential, $\kappa$ is a real constant and $\rho,g$ are real functions.
 Quasilinear Schr\"{o}dinger equations of the type \eqref{equ1-3}
with $\kappa>0$ arise in various branches of mathematical physics and have
been derived as models of several physical phenomena, such as superfluid
film equations in plasma physics \cite{Kuri} and the fluid mechanics
in condensed matter theory \cite{Bass, Kose, Quis, Take, Makh} and so on.
The related Schr\"{o}dinger equations for $\kappa=0$ have been extensively
studied (see e.g. \cite{Bere, Jean, Floe} and their references therein)
in the last few decades. For $\kappa>0$, the existence of a positive
ground state solution has been proved in \cite{Pop} by using a constrained
minimization argument, which gives a solution of \eqref{equ1-2} with
an unknown Lagrange multiplier $\lambda$ in front of nonlinear term.
In \cite{Liu1}, the authors establish the existence of ground states
 of soliton type solutions by a minimization argument. In \cite{Liu2},
by a change of variables the quasilinear problem was transformed to a
semilinear one and Orlicz space framework was used as the working space,
and they were able to prove the existence of positive solutions of
\eqref{equ1-2} by the mountain-pass theorem. The same method of change
of variables was used recently also in \cite{Coli}, but the usual Sobolev
space $H^{1}(\mathbb{R}^N)$ framework was used as the working space and
they studied different class of nonlinearities. In \cite{LWW}, it was
established the existence of both one-sign and nodal ground states of
soliton type solutions by the Nehari method. In \cite{ZTZ}, where the
potential $V(x)$ and $g$ is allowed to be sign-changing, $g$ is of
superlinear growth at infinity in $u$, the author obtain the existence
of infinitely many nontrivial solutions by using dual approach and symmetric
 mountain pass theorem.

Recently, there has been a lot of results on  existence and multiplicity for
problem \eqref{equ1-1}. The existence of nontrivial weak solutions of \eqref{equ1-1}
has been proved in \cite{Seve} by using minimax methods and method of
Changes of variable, where $V$ is a positive continuous potential bounded
away from zero. In \cite{Alves}, the authors use variational method together
with the Lusternick-Schnirelmann category theory to get the existence and
multiplicity of nontrivial weak solutions, where $V$ is also a positive
continuous potential bounded away from zero. In \cite{AF}, the authors
established the multiplicity of positive weak solutions through using minimax
methods, where the potential $V$ is of form $V(x)=\lambda A(x)+1$ and $A(x)$
is a nonnegative continuous function. The other related results can be seen
in \cite{AFS} and the references therein.

In the above mentioned paper, the potential $V$ is always assumed to be
positive or vanish at infinity except \cite{ZTZ}. In the present paper
 we shall consider problem \eqref{equ1-1} with non-constant and sign-changing
potential. We will investigate the existence of at least two solutions
and the existence of infinitely many nontrivial solutions of \eqref{equ1-1}
through using the Ekeland's variational principle, variant mountain pass
theorem and symmetric mountain pass theorem. Our main results improve
the corresponding theorems in \cite{ZTZ} in some sense.

For stating our main result, we make the following assumptions on the potential
function $V(x)$
\begin{itemize}
\item[(A1)]  $V \in C(\mathbb{R}^N)$ and $\inf_{x \in \mathbb{R}^N} V(x)>-\infty$,
and there exists a constant $d_{0}>0$ such that
\[
 \lim_{|y|\to\infty}\operatorname{meas}
 (\{x \in \mathbb{R}^N: |x-y|\leq d_{0},\;V(x)\leq M\})=0,\quad \forall M>0.
\]
\end{itemize}
Inspired by \cite{LCY,ZTZ}, we can find a constant $V_{0}>0$ such that
$\overline{V}(x)=V(x)+V_{0}\geq1$ for all $x\in\mathbb{R}^N$, and
let $\overline{g}(x,u)=g(x,u)+V_{0}|u|^{p-2}u$, for
all $(x,u)\in\mathbb{R}^N\times\mathbb{R}$. Then it is easy to show the
following Lemma.

\begin{lemma}\label{lem1-1}
Equation \eqref{equ1-1} is equivalent to the  problem
\begin{equation}\label{equ1-4}
-\Delta_{p} u-(\Delta_{p}u^{2})u+ \overline{V}(x)|u|^{p-2}u=\overline{g}(x,u),\quad
  x\in \mathbb{R}^N.
\end{equation}
\end{lemma}

In what follows, we impose some assumptions on $\overline{g}$ and its primitive
$\overline{G}(x,t)=\int_0^t\overline{g}(x,s){\rm d}s$ as follows:
\begin{itemize}
\item[(A2)]  $\overline{g}\in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$
and there exist constant $C> 0$ and $2p< q <2p^{\ast}$ such that
\[
|\overline{g}(x,u)| \leq C( |u|^{p-1} + |u|^{q-1}),\quad
 \forall (x,u) \in \mathbb{R}^N \times \mathbb{R};
\]

\item[(A3)]  $\lim_{|u| \to \infty} \overline{G}(x,u)/|u|^{2p}= +\infty$
uniformly in $x\in\mathbb{R}^N$, and there exists $r_{0}>0$, $\tau< p$ and
$C_{0}$ such that $\inf\overline{G}(x,u)\geq C_{0}|u|^{\tau}>0$,
for all $(x,u)\in\mathbb{R}^N\times\mathbb{R}$, $|u|\geq r_{0}$;

\item[(A4)]  $\widetilde{\overline{G}}(x,u)
=\frac{1}{2p} u \overline{g}(x,u)-\overline{G}(x,u)\geq0$,
There exist $C_{1}$ and $\sigma >\frac{2N}{N+p}$ such that
\[
\Big(\overline{G}(x,u)\Big)^{\sigma} \leq C_{1} |u|^{p\sigma}
\widetilde{\overline{G}}(x,u)\quad \text{for all }
 (x,u)\in\mathbb{R}^N\times\mathbb{R}, \; |u|\geq r_{0};
\]

\item[(A5)]  There exist $\mu>2p$ and $C_{2}>0$ such that
$\mu \overline{G}(x,u)\leq u\overline{g}(x,u)+C_{2}|u|^{p}$,
for all $(x,u)\in\mathbb{R}^N\times\mathbb{R}$;

\item[(A6)]  There exist $\mu>2p$ and $r_{1}>0$ such that
$\mu \overline{G}(x,u)\leq u\overline{g}(x,u)$, for all
$(x,u)\in\mathbb{R}^N\times\mathbb{R}$ with $|u|\geq r_{1}$;

\item[(A7)]  $\lim_{|u| \to 0} \frac{\overline{G}(x,u)}{|u|^{p}}= 0$
 uniformly in $x\in\mathbb{R}^N$;

\item[(A8)]  $\overline{g}(x,-u) =-\overline{g}(x,u)$ for all
$(x,u) \in \mathbb{R}^N \times \mathbb{R}$.

\end{itemize}

\begin{remark}\label{rem1-1} \rm
It follows from (A3) and (A4) that
\begin{equation}\label{equ1-5}
\widetilde{\overline{G}}(x,u)\geq \frac{1}{C_{1}}
\Big(\frac{\overline{G}(x,u)}{|u|^p}\Big)^{\sigma} \to \infty
\end{equation}
uniformly for $x\in\mathbb{R}^N$ as $|u|\to \infty$.
\end{remark}

Now, we state our main results.

\begin{theorem}\label{thm1-1}
Suppose that conditions {\rm (A1)--(A4)} are satisfied. Then  \eqref{equ1-1}
possesses at least two solutions.
\end{theorem}

\begin{theorem} \label{thm1-2}
Suppose that conditions {\rm (A1)--(A3), (A5)} are satisfied.
Then  \eqref{equ1-1} possesses at least two solutions.
\end{theorem}

From (A2) and (A6), it is easy to verified that (A5) holds.
Thus we have the following corollary.

\begin{corollary}\label{cor1-1}
Suppose that conditions {\rm (A1)--(A3), (A6)} are satisfied.
Then  \eqref{equ1-1} possesses at least two solutions.
\end{corollary}

If we add the hypothesis (A8), we can obtain the infinitely many solutions
for problem \eqref{equ1-1}.

\begin{theorem}\label{thm1-3}
Assume that {\rm (A1)--(A4), (A8)} are satisfied. Then  \eqref{equ1-1}
possesses infinitely many nontrivial solutions.
\end{theorem}

\begin{theorem}\label{thm1-4}
Assume that {\rm (A1)--(A3), (A5), (A7), (A8)} are satisfied. Then \eqref{equ1-1}
 possesses infinitely many nontrivial solutions.
\end{theorem}

\begin{corollary}\label{cor1-2}
Assume that {\rm (A1)--(A3), (A6)--(A8)} are satisfied. Then \eqref{equ1-1}
possesses infinitely many nontrivial solutions.
\end{corollary}

\begin{remark}\label{rem1-2} \rm
If we use the following assumption instead of (A2):
\begin{itemize}
\item[(A2')]  $\overline{g}\in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$
and there exist constant $C_{3}> 0$, $p<r\leq 2p$ and $2p< q <2p^{\ast}$ such that
\[
|\overline{g}(x,u)| \leq C_{3}( |u|^{r-1} + |u|^{q-1}),\,\, \forall (x,u) \in \mathbb{R}^N \times \mathbb{R}.
\]
\end{itemize}
Then the assumption (A7) is not needed.
Thus we can get the similar results as Theorem \ref{thm1-1}--\ref{thm1-4}.
Here we omit their statements.
\end{remark}


\section{Variational setting and preliminary results}\label{1}

As usual, for $1\leq s \leq +\infty$, we let
\[
\|u\|_{s}= (\int_{\mathbb{R}^N}|u(x)|^{s})^{1/s},
u \in L^{s}(\mathbb{R}^N).
\]
 We denote $C$, $C_{i}$  $(i=0,1,2,\cdots)$ as the various positive constants
 throughout this paper. Throughout this section, we make the following
assumption on $\overline{V}$ instead of (A1):
\begin{itemize}
\item[(A1')]  $\overline{V}\in C(\mathbb{R}^N,\mathbb{R})$,
 and $\inf_{x \in \mathbb{R}^N}\overline{V}(x) >0$, and there exists
a constant $d_{0}>0$ such that
\[
 \lim_{|y|\to\infty}\operatorname{meas} (\{x \in \mathbb{R}^N:
|x-y|\leq d_{0},\;\overline{V}(x)\leq M\})=0,\quad \forall M>0.
\]
\end{itemize}
Let
\[
E:= \{u \in W^{1,p}(\mathbb{R}^N, \mathbb{R}):
\int_{\mathbb{R}^N} \overline{V}(x)|u|^{p}{\rm d}x  < \infty \},
\]
which is endowed with the norm
\[
\|u\|_{E}= \Big(\int_{\mathbb{R}^N}(|\nabla u|^p
+\overline{V}(x)|u|^p)\,{\rm d}x\Big)^{1/p}.
\]

Under assumption (A1'), the embedding $E \hookrightarrow L^{s}(\mathbb{R}^N)$
is continuous for $s \in [p, p^{\ast})$, and
$E \hookrightarrow L_{loc}^{s}(\mathbb{R}^N)$ is compact for $s \in [p, p^{\ast})$,
i.e., there are constants $a_{s} > 0$ such that
\[
\|u\|_{s} \leq a_{s}\|u\|_{E},\quad \forall u \in E,\; s \in [p, p^{\ast}).
\]
Furthermore, under  assumption (A1'), we have the following compactness
embedding lemma due to \cite{Bartsch, Bw, ZS}.

\begin{lemma}\label{lem2-1}
Under assumption {\rm (A1')}, the embedding from $E$ into $L^{s}(\mathbb{R}^N)$
is compact for $p\leq s < p^{\ast}$.
\end{lemma}

The energy functional $J : E \to \mathbb{R}$ formally can be given by
\begin{align*}
J(u)&=\frac{1}{p} \int_{\mathbb{R}^N} |\nabla u|^{p}{\rm d}x  
 + \frac{2^{p-1}}{p} \int_{\mathbb{R}^N} |\nabla u|^{p}|u|^{p}{\rm d}x 
 + \frac{1}{p} \int_{\mathbb{R}^N} \overline{V}(x)|u|^{p}{\rm d}x \\
&\quad - \int_{\mathbb{R}^N}\overline{G}(x,u){\rm d}x \\
&=\frac{1}{p} \int_{\mathbb{R}^N} (1+2^{p-1}|u|^{p})|\nabla u|^{p}{\rm d}x 
 + \frac{1}{p} \int_{\mathbb{R}^N} \overline{V}(x)|u|^{p}{\rm d}x  
 - \int_{\mathbb{R}^N}\overline{G}(x,u){\rm d}x.
\end{align*}
Since the integral $\int_{\mathbb{R}^N}|\nabla u|^{p}|u|^{p}{\rm d}x$ may
 be infinity, $J$ is not well defined in general in $E$. To overcome this 
difficulty, we apply an argument developed by \cite{Liu2}. We make the 
change of variables by $v= f^{-1}(u)$, where $f$ is defined by
\[
f'(t)=  \frac{1}{[1 + 2^{p-1}|f(t)|^{p} ]^{1/p}},\quad t\in[0,\infty),
\]
and
\[
f(-t)=-f(t),\,\,  \text{t}\in(-\infty,0].
\]
Some properties of the function $f$ are listed as follows.

\begin{lemma}\label{lem2-2}
Concerning the function $f(t)$ and its derivative satisfy the following properties:
\begin{enumerate}
\item   $f$ is uniquely defined, $C^{2}$ and invertible;
\item  $|f'(t)|\leq 1$ for all $t \in \mathbb{R}$;
\item  $|f(t)|\leq |t|$ for all $t \in \mathbb{R}$;
\item  $\frac{f(t)}{t}\to 1$ as $t\to 0$;
\item  $\frac{f(t)}{\sqrt{t}}\to a>0$ as $t\to +\infty$;
\item  $\frac{f(t)}{2} \leq tf'(t) \leq f(t)$ for all $t> 0$;
\item  $\frac{f^{2}(t)}{2} \leq tf'(t)f(t) \leq f^{2}(t)$ for all $t \in \mathbb{R}$;
\item  $|f(t)|\leq 2^{\frac{1}{2p}} |t|^{\frac{1}{2}}$ for all $t \in \mathbb{R}$;
\item  there exists a positive constant $C_{4}$ such that
\[
|f(t)|\geq \begin{cases}
C_{4}|t|,& |t|\leq 1,\\
C_{4}|t|^{\frac{1}{2}},& |t|\geq 1;
\end{cases}
\]

\item
\[
f^{2}(st)\leq  \begin{cases}
sf^{2}(t),& 0\leq s \leq 1,\\
 s^{2}f^{2}(t),& s \geq 1;
\end{cases}
\]

\item  $|f(t)f'(t)|\leq \frac{1}{2^{\frac{p-1}{p}}}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We only prove properties (10). Since the function $(f^{2})''> 0$, 
in $[0, +\infty)$, and therefore item $f^{2}$ is strictly convex,
\[
f^{2}((1-s)0 +s t) \leq (1-s)f^{2}(0) + sf^{2}(t) = sf^{2}(t).
\]
In order to prove $f^{2}(st)\leq s^{2}f^{2}(t)$, when $s \geq 1$.
We notice that, since $f''\leq 0$ in $[0, +\infty)$, we have that
$f'$ is non-increasing in this interval. For any $t\geq 0$ fixed we
consider the function $h(s):= f(st)- sf(t)$ defined for $s \geq 1$.
We have that $h'(s):= tf'(st)- f(t)\leq tf'(t)- f(t)\leq 0$, by $(f_{6})$.
Since $h(1) = 0$ we consider that $h(s)\leq 0$ for any $s\geq 1$;
 that is, $f(st)\leq sf(t)$ for any $t\geq 0$ and $s \geq 1$.
Thus the proof is complete.
\end{proof}

By the change of variables, from $J(u)$ we can define the following functional
\begin{equation}\label{equ2-1}
\begin{split}
I(v)=\frac{1}{p} \int_{\mathbb{R}^N}(|\nabla v|^{p} 
 + \overline{V}(x)|f(v)|^{p}){\rm d}x 
 - \int_{\mathbb{R}^N}\overline{G}(x,f(v)){\rm d}x,
\end{split}
\end{equation}
which is well defined on the space $E$. From (A2), we have
\[
\overline{G}(x,u)\leq C(|u|^p+|u|^q),\quad \text{for all }
 (x,u)\in\mathbb{R}^N\times\mathbb{R}.
\]
By standard arguments, it is easy to show that $I\in C^1(E,\mathbb{R})$, and
\begin{equation}\label{equ2-2}
\begin{split}
\langle I'(v),w \rangle
&= \int_{\mathbb{R}^N}|\nabla v|^{p-2} \nabla v \nabla w{\rm d}x
 + \int_{\mathbb{R}^N} \overline{V}(x)|f(v)|^{p-2}f(v)f'(v) w {\rm d}x \\
&\quad -\int_{\mathbb{R}^N} \overline{g}(x,f(v))f'(v) w {\rm d}x,
\end{split}
\end{equation}
for any $w\in E$. 
Moreover, the critical points of $I$ are the weak solutions of the
following equation
\[
-\Delta_p v+ \overline{V}(x)|f(v)|^{p-2}f(v)f'(v)=\overline{g}(x,f(v))f'(v).
\]
We also observe that if $v$ is a critical point of the functional $I$,
then $u= f(v)$ is a critical point of the functional $J$, i.e. $u= f(v)$
is a solution of problem \eqref{equ1-4}.

Next, we present the relationship between the norm $\|u\|_{E}$ in $E$ 
and $\int_{\mathbb{R}^N}(|\nabla u|^p+\overline{V}(x)|f(u)|^p){\rm d}x$.

\begin{proposition}\label{pro2-1}
There exist two constants $C_{5}>0$ and $\rho>0$ such that
\[
\int_{\mathbb{R}^N}(|\nabla u|^p+\overline{V}(x)|f(u)|^p){\rm d}x
\geq C_{5}\|u\|^{p}_{E},\quad \forall\,\, u\in\{u\in E:\|u\|_{E}\leq\rho\}.
\]
\end{proposition}

\begin{proof}
Suppose by contradiction, there exists a sequence $\{u_{n}\}\subset E$ 
verifying $u_{n}\neq0$, for all $n\in\mathbb{N}$ and $\|u_{n}\|_{E}\to 0$, such that
\begin{equation}\label{equ2-3}
\int_{\mathbb{R}^N}\Big(\frac{|\nabla u_{n}|^{p}}{\|u_{n}\|^{p}_{E}}
+ \overline{V}(x)\frac{|f(u_{n})|^{p}}{\|u_{n}\|^{p}_{E}}\Big){\rm d}x \to 0.
\end{equation}
Set $v_{n}= u_{n}/\|u_{n}\|_{E}$, then $\|v_{n}\|_{E}= 1$, passing to a 
subsequence, by Lemma \ref{lem2-1}, we may assume that $v_{n}\rightharpoonup v$ 
in $E$, $v_{n}\to v$ in $L^{s}(\mathbb{R}^N)$ for $s\in[p,p^{\ast})$, 
$v_{n}\to v$ a.e $\mathbb{R}^N$. Therefore, \eqref{equ2-3} implies that
\begin{equation}\label{equ2-4}
\int_{\mathbb{R}^N}|\nabla v_{n}|^{p}{\rm d}x  \to 0,\quad
\int_{\mathbb{R}^N} \overline{V}(x)\frac{|f(u_{n})|^{p}}{\|u_{n}\|^{p}_{E}}{\rm d}x 
 \to 0, \quad
\int_{\mathbb{R}^N} \overline{V}(x)|v_{n}|^{p}{\rm d}x  \to 1.
\end{equation}
Similar to the idea in \cite{Wu}, we assert that for each $\varepsilon > 0$, 
there exists $C_{6}> 0$ independent of $n$ such that
 $\operatorname{meas}(\Omega_{n})< \varepsilon$, where
 $\Omega_{n}:= \{x\in \mathbb{R}^N: |u_{n}(x)|\geq C_{6}\}$. 
Otherwise, there is an $\varepsilon_{0} > 0$ and a subsequence 
$\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that for any positive integer $k$,
\[
\operatorname{meas}(\{x\in \mathbb{R}^N: |u_{n_{k}}(x)|\geq k\})
\geq \varepsilon_{0} > 0.
\]

Set $\Omega_{n_{k}}:= \{x\in \mathbb{R}^N: |u_{n_{k}}(x)|\geq k\}$. 
By (3) and (9) of Lemma \ref{lem2-2}, we have
\begin{align*}
\|u_{n_{k}}\|_E^p
&\geq\int_{\mathbb{R}^N} \overline{V}(x)|u_{n_{k}}|^p{\rm d}x
 \geq  \int_{\mathbb{R}^N} \overline{V}(x)|f(u_{n_{k}})|^{p}{\rm d}x\\
&\geq \int_{\Omega_{n_{k}}} \overline{V}(x)|f(u_{n_{k}})|^{p}{\rm d}x
 \geq C_{6}k^{\frac{p}{2}}\varepsilon_{0},\\
\end{align*}
which implies a contradiction. Hence the assertion is true.

On the one hand, by  the absolutely continuity of Lebesgue integral, 
there exists $\delta>0$ such that when $A\subset\mathbb{R}^N$ with 
$\operatorname{meas}\, (A)<\delta$, we have
\[
\int_{A}\overline{V}(x)|v_{n}(x)|^p{\rm d}x<\frac{1}{p}.
\]

Hence, we can find a constant $C_{7}>0$ such that 
$\operatorname{meas}\, (\Omega_n)<\delta$. Thus we infer that
\begin{equation}\label{equ2-5}
\int_{\Omega_n} \overline{V}(x)|v_{n}(x)|^{p}\,{\rm d}x\leq\frac{1}{p}.
\end{equation}
On the other hand, when $|u_{n}(x)|\leq C_{6}$, by (9) and (10)
 of Lemma \ref{lem2-2}, we have
\begin{equation}\label{equ2-6}
\int_{\mathbb{R}^N \backslash \Omega_{n}} \overline{V}(x)|v_{n}|^{p} {\rm d}x
=\int_{\mathbb{R}^N \backslash \Omega_{n}} \overline{V}(x)
\frac{|u_{n}|^{p}}{\|u_{n}\|^{p}_{E}} {\rm d}x 
\leq C_{7}\int_{\mathbb{R}^N \backslash \Omega_{n}} \overline{V}(x)
\frac{|f(u_{n})|^{p}}{\|u_{n}\|^{p}_{E}}{\rm d}x  \to 0.
\end{equation}
Combining \eqref{equ2-5} and \eqref{equ2-6}, we have
\[
\int_{\mathbb{R}^N}  \overline{V}(x)|v_{n}(x)|^{p}
= \int_{\mathbb{R}^N\backslash \Omega_{n}}  \overline{V}(x)|v_{n}(x)|^{p}
+ \int_{\Omega_{n}} \overline{V}(x)|v_{n}(x)|^{p}\leq \frac{1}{p}+o(1),
\]
which implies that $1\leq \frac{1}{p}$, a contradiction. The proof is complete.
\end{proof}

\begin{proposition}\label{pro2-2}
For any sequence $\{u_{n}\}\subset E$ satisfying
\[
\int_{\mathbb{R}^N}(|\nabla u_{n}|^{p}+\overline{V}(x)|f(u_{n})|^{p}){\rm d}x
\leq C_{8},
\]
there exists a constant $C_{9}>0$ such that
\[
\int_{\mathbb{R}^N}(|\nabla u_{n}|^{p}+\overline{V}(x)|f(u_{n})|^{p}){\rm d}x
\geq C_{9}\|u_{n}\|^{p}_{E},\quad \forall\, n\in\mathbb{N}.
\]
\end{proposition}

\begin{proof}
We argue by conradiction, so there exists a subsequence 
$\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that
\[
\int_{\mathbb{R}^N}\Big(\frac{|\nabla u_{n_{k}}|^{p}}{\|u_{n_{k}}\|^{p}_{E}}
+\overline{V}(x)\frac{|f(u_{n_{k}})|^p}{\|u_{n_{k}}\|^{p}_{E}}\Big){\rm d}x\to 0
\quad \text{as }k\to\infty.
\]
The rest of the proof is similar to Proposition \ref{pro2-1},
we can deduce the conclusion.
\end{proof}

At the end of this section, we recall the variant mountain pass theorem 
and symmetric mountain pass theorem which are used to prove our main result.

\begin{theorem}[\cite{Sun}]\label{thm2-1}
 Let $E$ be a real Banach space with its dual space $E^{\ast}$, 
and suppose that $I\in C^{1}(E,\mathbb{R})$ satisfies
\[
\max\{I(0), I(e)\}\leq \mu< \eta\leq \inf_{\|u\|= \rho}I(u),
\]
for some $\rho> 0$ and $e\in E$ with $\|e\|> \rho$.
Let $c\geq \eta$ be characterized by
\[
c= \inf_{\gamma\in \Gamma}\max_{0\leq \tau\leq 1}I(\gamma(\tau)),
\]
where $\Gamma= \{\gamma\in C([0,1], E): \gamma(0)= 0, \gamma(1)= e\}$
is the set of continuous paths joining $0$ and $e$, then there exists
a sequence $\{u_{n}\}\subset E$ such that
\begin{equation}\label{equ2-7}
I(u_{n})\to c\geq \eta\quad \text{and}\quad
 (1+ \|u_{n}\|)\|I'(u_{n})\|_{E^{\ast}}\to 0,\quad \text{as } n\to\infty.
\end{equation}
\end{theorem}

A sequence $\{v_{n}\}\subset E$ is said to be a Cerami sequence 
(simply $(C)_{c}$) if $I(v_{n})\to c$ and $(1+\|v_{n}\|_{E})I'(v_{n})\to 0$, 
$I$ is said to satisfy the $(C)_{c}$ condition if any $(C)_{c}$ sequence 
has a convergent subsequence.

\begin{theorem}[\cite{Rabi}]\label{thm2-2}
Let $E$ be an infinite dimensional Banach space, $E= Y\oplus Z$, 
where $Y$ is finite dimensional. If $\varphi \in C^{1}(E,\mathbb{R})$
 satisfies $(C)_{c}$-condition for all $c> 0$, and
\begin{enumerate}
\item  $\varphi (0)= 0$, $\varphi (-u)= \varphi (u)$ for all $u\in E$;
\item  there exist constants $\rho$, $\alpha$ such that
 $\varphi|\partial B_{\rho} \cap Z \geq \alpha$;
\item  for any finite dimensional subspace $\widetilde{E} \subset E$, 
there is $R= R(\widetilde{E} )> 0$ such that $\varphi (u)\leq 0$ 
on $\widetilde{E}\setminus B_{R}$.
\end{enumerate}
Then $\varphi$ possesses an unbounded sequence of critical values.
\end{theorem}


\section{$(C)_c$ condition}\label{2}

In this section, we will prove the bondedness of $(C)_c$ sequence and 
then show that bounded $(C)_c$ sequence is strongly convergence in $E$.

\begin{lemma}\label{lem3-1}
Any bounded $(C)_c$ sequence of $I$ possesses a convergence subsequence in $E$.
\end{lemma}

\begin{proof}
 Assume that $\{v_{n}\}\subset E$ is a bounded sequence satisfying
\begin{equation}\label{equ3-1}
I(v_{n})\to c\quad \text{and}\quad (1+ \|v_{n}\|_{E})I'(v_{n})\to 0.
\end{equation}
Going if necessary to a subsequence, we can assume that 
$v_{n}\rightharpoonup v$ in $E$. By Lemma \ref{lem2-1}, 
$v_{n}\to v$ in $L^{s}(\mathbb{R}^N)$ for all 
$p\leq s < p^{\ast}$ and $v_{n}\to v$ a.e. on $\mathbb{R}^N$. 
First, we claim that there exists $C_{10}> 0$ such that
\begin{equation}\label{equ3-2}
\begin{split}
&\int_{\mathbb{R}^N} |\nabla(v_{n}-v)|^{p} 
 + \overline{V}(x)\Big(|f(v_{n})|^{p-2} f(v_{n}) f'(v_{n}) \\
&\quad - |f(v)|^{p-2} f(v) f'(v)\Big) (v_{n}-v){\rm d}x \\
&\geq \int_{\mathbb{R}^N} (|\nabla v_{n}|^{p-1}- |\nabla v|^{p-1}) \nabla (v_{n}-v)+\overline{V}(x)\Big(|f(v_{n})|^{p-2} f(v_{n}) f'(v_{n}) \\
&\quad -|f(v)|^{p-2} f(v) f'(v)\Big) (v_{n}-v){\rm d}x \\
&\geq C_{10}\|v_{n}-v \|^{p}_{E}.
\end{split}
\end{equation}
Indeed, we may assume that $v_{n}\neq v$. Set
\[
w_{n}=\frac{v_{n}-v}{\|v_{n}-v \|_{E}},\quad
h_{n}=\frac{|f(v_{n})|^{p-2} f(v_{n}) f'(v_{n})
 - |f(v)|^{p-2} f(v) f'(v)}{|v_{n}-v|^{p-1}}.
\]
We argue by contradiction and assume that
\[
\int_{\mathbb{R}^N} |\nabla w_{n}|^{p}
+ \overline{V}(x) h_{n}(x) w^{p}_{n}{\rm d}x \to 0.
\]
Since
\[
\frac{d}{dt}\Big(|f(t)|^{p-2} f(t) f'(t)\Big)
= |f(t)|^{p-2} |f'(t)|^{2}\Big[p- 1- \frac{2^{p-1} |f(t)|^{p}}{1+2^{p-1}|f(t)|^{p} }
 \Big] > 0,
\]
so, $|f(t)|^{p-2} f(t) f'(t)$ is strictly increasing and for each
$C_{11}> 0$ there is $\delta_{1}> 0$ such that
\[
\frac{d}{dt}\Big(|f(t)|^{p-2} f(t) f'(t)\Big) \geq \delta_{1}\quad \text{as }
 |t|\leq C_{11}.
\]
From this, we can see that $h_{n}(x)$ is positive. Hence
\[
\int_{\mathbb{R}^N}|\nabla w_{n}|^{p}{\rm d}x  \to 0,\quad
 \int_{\mathbb{R}^N} \overline{V}(x)h_{n}(x) w^{p}_{n}{\rm d}x  \to 0, \quad
\int_{\mathbb{R}^N} \overline{V}(x)|w_{n}|^{p}{\rm d}x  \to 1.
\]
By a similar argument as Proposition \ref{pro2-1}, we can conclude
a contradiction.

On the other hand, by (2), (3), (8) and (11) of Lemma \ref{thm2-2}, 
(A2) and the definition of the $f'(t)$, we have
\begin{align*} %\label{equ3-3}
&\big|\int_{\mathbb{R}^N} \Big(\overline{g}(x,f(v_{n})) f'(v_{n})
 - \overline{g}(x,f(v)) f'(v)\Big) (v_{n}-v){\rm d}x \big|\\
&\leq \Big(\int_{\mathbb{R}^N} |\overline{g}(x,f(v_{n})) f'(v_{n})| 
 + \int_{\mathbb{R}^N} |\overline{g}(x,f(v)) f'(v)|\Big) |v_{n}-v|{\rm d}x \\
&\leq \int_{\mathbb{R}^N} C_{12}\big(|f(v_{n})|^{p-1}+ |f(v_{n})|^{q-1}\big)
 |f'(v_{n})||v_{n}-v|{\rm d}x\\
&\quad+ \int_{\mathbb{R}^N} C_{12}\Big(|f(v)|^{p-1} + |f(v)|^{q-1}\Big)|f'(v)|
 |v_{n}-v|{\rm d}x\\
&\leq \int_{\mathbb{R}^N} C_{12}\Big(|f(v_{n})|^{p-1}|f'(v_{n})|
 + |f(v_{n})|^{q-1}|f'(v_{n})|\Big)|v_{n}-v|{\rm d}x\\
&\quad +\int_{\mathbb{R}^N} C_{12}\Big(|f(v)|^{p-1}|f'(v)|
 +|f(v)|^{q-1}|f'(v)|\Big)|v_{n}-v|{\rm d}x\\
&\leq \int_{\mathbb{R}^N} C_{12}\Big(|f(v_{n})|^{p-1}
 + \frac{|f(v_{n})|^{q-1}}{[1 + 2^{p-1}|f(v_{n})|^{p} ]^{1/p}}\Big)|v_{n}-v|{\rm d}x\\
&\quad +\int_{\mathbb{R}^N} C_{12}\Big(|f(v)|^{p-1}
 + \frac{|f(v)|^{q-1}}{[1 + 2^{p-1}|f(v)|^{p} ]^{1/p}}\Big)|v_{n}-v|{\rm d}x\\
&\leq \int_{\mathbb{R}^N} C_{12}\Big(|f(v_{n})|^{p-1}+ |f(v_{n})|^{q-2}
 + |f(v)|^{p-1}+ |f(v)|^{q-2}\Big)|v_{n}-v|{\rm d}x\\
&\leq \int_{\mathbb{R}^N} C_{12}\Big(|v_{n}|^{p-1} +|v_{n}|^{\frac{q}{2}-1}
 + |v|^{p-1} + |v|^{\frac{q}{2}-1}\Big) |v_{n}-v|{\rm d}x \\
&\leq C_{12}\Big((\|v_{n}\|^{p-1}_{p}+ \|v\|^{p-1}_{p}) \|v_{n}- v\|_{p}\Big)
 + C_{12}\Big((\|v_{n}\|^{\frac{q-2}{2}}_{\frac{q}{2}}
 + \|v\|^{\frac{q-2}{2}}_{\frac{q}{2}}) \|v_{n}- v\|_{\frac{q}{2}}\Big)\\
&= o(1).
\end{align*}
Therefore, by \eqref{equ3-2} and  the above inequality,  we have
\begin{align*}
o(1)&= \langle I'(v_{n})- I'(v), v_{n}-v\rangle\\
&= \int_{\mathbb{R}^N} \Big[|\nabla(v_{n}-v)|^{p} 
 + \overline{V}(x)\Big(|f(v_{n})|^{p-2} f(v_{n}) f'(v_{n}) \\
&\quad - |f(v)|^{p-2} f(v) f'(v)\Big) (v_{n}-v)\Big]{\rm d}x \\
&\quad - \int_{\mathbb{R}^N} \Big(\overline{g}(x,f(v_{n})) f'(v_{n})
 - \overline{g}(x,f(v)) f'(v)\Big) (v_{n}-v){\rm d}x \\
&\geq C_{13}\|v_{n}-v \|^{p}_{E}+ o(1),
\end{align*}
which implies that $\|v_{n}-v \|_{E}\to 0$ as $n\to \infty$. 
The proof is complete.
\end{proof}

\begin{lemma}\label{lem3-2}
Suppose that {\rm (A1'), (A2)-(A4)} are satisfied. 
Then any $(C)_{c}$ sequence of $I$ is bounded in $E$.
\end{lemma}

\begin{proof}
Let $\{v_{n}\}\subset E$ be such that
\begin{equation}\label{equ3-4}
I(v_{n})\to c\quad \text{and}\quad  (1+ \|v_{n}\|_{E})I'(v_{n})\to 0.
\end{equation}
Thus, there is a constant $C_{14}> 0$ such that
\begin{equation}\label{equ3-5}
I(v_{n})-\frac{1}{2p}\langle I'(v_{n}),v_{n}\rangle\leq C_{14}.
\end{equation}
Firstly, we prove that there exists $C_{15}> 0$ independent of $n$ such that
\begin{equation}\label{equ3-6}
\int_{\mathbb{R}^N}\Big(|\nabla v_{n}|^{p}+ \overline{V}(x)|f(v_{n})|^{p}\Big)
{\rm d}x  \leq C_{15}.
\end{equation}
Suppose by contradiction that
\[
\|v_{n}\|^{p}_{0}:= \int_{\mathbb{R}^N}\Big(|\nabla v_{n}|^{p}
+ \overline{V}(x)|f(v_{n})|^{p}\Big){\rm d}x \to \infty\quad
 \text{as } n\to\infty.
\]
Setting $\widetilde{f}(v_{n}):= f(v_{n})/\|v_{n}\|_{0}$, then
$\|\widetilde{f}(v_{n})\|_{E}\leq 1$. Passing to a subsequence, we may assume
 that $\widetilde{f}(v_{n})\rightharpoonup w$ in $E$,
$\widetilde{f}(v_{n})\to w$ in $L^{s}(\mathbb{R}^N)$, $p \leq s < p^{\ast}$,
and $\widetilde{f}(v_{n})\to w$ a.e. $\mathbb{R}^N$. It follows from
\eqref{equ3-4} that
\begin{equation}\label{equ3-7}
\lim_{n\to \infty} \int_{\mathbb{R}^N}
\frac{|\overline{G}(x,f(v_{n}))|}{\|v_{n}\|^{p}_{0}}{\rm d}x \geq \frac{1}{p}.
\end{equation}
Let $\varphi_{n}=f(v_{n})/f'(v_{n})$, by \eqref{equ3-5}, we have
\begin{align*}
C_{14}
&\geq I(v_{n})-\frac{1}{2p}\langle I'(v_{n}),\varphi_{n}\rangle\\
&=\frac{1}{2p} \int_{\mathbb{R}^N}|\nabla v_{n}|^{p} |f'(v_{n})|^{p}{\rm d}x
 + \frac{1}{2p} \int_{\mathbb{R}^N} \overline{V}(x)|f(v_{n})|^{p}{\rm d}x\\
&\quad  + \int_{\mathbb{R}^N}\frac{1}{2p} \overline{g}(x,f(v_{n}))f(v_{n}){\rm d}x
-\int_{\mathbb{R}^N} \overline{G}(x,f(v_{n})){\rm d}x, \\
\end{align*}
which implies
\begin{equation}\label{equ3-8}
C_{14}\geq \int_{\mathbb{R}^N} \widetilde{\overline{G}}(x,f(v_{n})){\rm d}x.
\end{equation}
Set
\[
h(r):= \inf\{ \widetilde{\overline{G}}(x,f(v_{n})): x\in \mathbb{R}^N ,
|f(v_{n})|\geq r\}\quad r\geq 0.
\]
By \eqref{equ1-5}, $h(r)\to \infty$ as $r\to \infty$. For $0\leq a< b$, let
$\Omega_{n}(a,b)= \{x\in \mathbb{R}^N:a\leq |f(v_{n}(x))|< b\}$.
Hence, it follows from \eqref{equ3-8} that
\begin{align*}
C_{14}
&\geq  \int_{\Omega_{n}(0,r)} \widetilde{\overline{G}}(x,f(v_{n}))
 +  \int_{\Omega_{n}(r, +\infty)} \widetilde{\overline{G}}(x,f(v_{n}))\\
&\geq \int_{\Omega_{n}(0,r)} \widetilde{\overline{G}}(x,f(v_{n}))
 + h(r) \operatorname{meas}(\Omega_{n}(r, +\infty)),
\end{align*}
which implies that $ \operatorname{meas}(\Omega_{n}(r, +\infty)) \to 0$
as $r\to \infty$ uniformly in $n$. Thus, for any $s\in [p,2p^{\ast})$,
by (8) of Lemma \ref{lem2-2}, H\"{o}lder inequality and Sobolev embedding, we have
\begin{equation}\label{equ3-9}
\begin{split}
&\int_{\Omega_{n}(r, +\infty)} \widetilde{f}^{s}(v_{n}){\rm d}x \\
&\leq \Big(\int_{\Omega_{n}(r, +\infty)} \widetilde{f}^{2p^{\ast}}(v_{n}){\rm d}x
 \Big)^{\frac{s}{2p^{\ast}}}
 \Big(\operatorname{meas}(\Omega_{n}(r, +\infty))\Big)^{\frac{2p^{\ast}-s}{2p^{\ast}}}\\
& \leq \frac{C_{16}}{\|v_{n}\|^{s}_{0}}\Big(\int_{\Omega_{n}(r, +\infty)}
 |\nabla f^{2}(v_{n})|^{p}\Big)^{\frac{s}{2p}}
 \Big(\operatorname{meas}(\Omega_{n}(r, +\infty))\Big)^{\frac{2p^{\ast}-s}{2p^{\ast}}}\\
&\leq \frac{C_{17}}{\|v_{n}\|^{s}_{0}}
 \Big(\int_{\Omega_{n}(r, +\infty)} |\nabla v_{n}|^{p}\Big)^{\frac{s}{2p}}
 \Big(\operatorname{meas}(\Omega_{n}(r, +\infty))\Big)^{\frac{2p^{\ast}-s}{2p^{\ast}}}\\
&\leq C_{17}\|v_{n}\|^{-\frac{s}{2}}_{0}
 \Big(\operatorname{meas}(\Omega_{n}(r, +\infty))\Big)^{\frac{2p^{\ast}-s}{2p^{\ast}}}
 \to 0,
\end{split}
\end{equation}
as $r\to \infty$ uniformly in $n$.

If $w=0$, then $\widetilde{f}(v_{n})=\frac{f(v_{n})}{\|v_{n}\|_{0}}\to 0$ in 
$L^{s}(\mathbb{R}^N)$, $p\leq s< p^{\ast}$. For any $0< \epsilon < \frac{1}{4p}$, 
there exist large $r_{1}$, $N_{0}\in\mathbb{N}$ such that
\begin{equation}\label{equ3-10}
\begin{split}
&\int_{\Omega_{n}(0, r_{1})}\frac{|\overline{G}(x, f(v_{n}))|}{|f(v_{n})|^{p}}
 |\widetilde{f}(v_{n})|^{p}{\rm d}x\\
&\leq \int_{\Omega_{n}(0, r_{1})}\frac{C_{18}|f(v_{n})|^{p}
 + C_{19}|f(v_{n})|^{q}}{|f(v_{n})|^{p}}|\widetilde{f}(v_{n})|^{p}{\rm d}x \\
&\leq (C_{18}+ C_{19} r^{q-p}_{1}) \int_{\Omega_{n}(0, r_{1})} 
 |\widetilde{f}(v_{n})|^{p}{\rm d}x\\
&\leq (C_{18}+ C_{19} r^{q-p}_{1}) 
 \int_{\mathbb{R}^N} |\widetilde{f}(v_{n})|^{p}{\rm d}x< \epsilon,
\end{split}
\end{equation}
for all $n> N_{0}$. Set $\sigma'= \frac{\sigma}{\sigma-1}$. 
Since $\sigma > \frac{2N}{N+p}$, so $p\sigma'\in (p,2p^{\ast})$. 
Hence, it follows from (A4) and \eqref{equ3-8} that
\begin{equation}\label{equ3-11}
\begin{split}
&\int_{\Omega_{n}(r_{1}, +\infty)} 
 \frac{|\overline{G}(x, f(v_{n}))|}{|f(v_{n})|^{p}}|\widetilde{f}(v_{n})|^{p}{\rm d}x  \\
&\leq \Big(\int_{\Omega_{n}(r_{1},+\infty)}
 (\frac{|\overline{G}(x, f(v_{n}))|}{|f(v_{n})|^p})^{\sigma}{\rm d}x 
 \Big)^{1/\sigma} 
  \Big(\int_{\Omega_{n}(r_{1}, +\infty)}|\widetilde{f}(v_{n})|^{p\sigma'}{\rm d}x
  \Big)^{1/\sigma'}\\
& \leq C^{1/\sigma}_{20} \Big(\int_{\Omega_{n}(r_{1}, +\infty)}
 \widetilde{\overline{G}}(x,f(v_{n}){\rm d}x \Big)^{1/\sigma}
 \Big(\int_{\Omega_{n}(r_{1}, +\infty)}|\widetilde{f}(v_{n})
 |^{p\sigma'}{\rm d}x \Big)^{1/\sigma'}\\
& \leq  C_{21}\Big(\int_{\Omega_{n}(r_{1}, +\infty)}|\widetilde{f}(v_{n})
 |^{p\sigma'}{\rm d}x \Big)^{1/\sigma'} < \epsilon,
\end{split}
\end{equation}
for all $n$. Combining \eqref{equ3-10} with \eqref{equ3-11}, we have
\[
\int_{\mathbb{R}^N}\frac{\overline{G}(x,f(v_{n}))}{\|v_{n}\|^{p}_{0}}{\rm d}x
 =\Big(\int_{\Omega_{n}(0,r_{1})}+ \int_{\Omega_{n}(r_{1}, +\infty)}\Big)
\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{p}}|\widetilde{f}(v_{n})|^{p}{\rm d}x
< 2\epsilon <\frac{1}{p},
\]
for all $n> N_{0}$, which contradicts \eqref{equ3-7}.

If $w\neq 0$, then $\operatorname{meas}(\Omega)> 0$, where
 $\Omega := \{x\in \mathbb{R}^N : w\neq 0 \}$. For $x\in \Omega$, 
$|f(v_{n})|\to \infty$ as $n\to \infty$. Hence 
$\Omega \subset \Omega_{n}(r_{0},\infty)$ for large $n\in N$, where 
$r_{0}$ is given in $(A3)$. By $(A3)$, we have
\[
\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}\to
+\infty\quad \text{as } n\to\infty.
\]
Hence, using Fatou's lemma, we have
\begin{equation}\label{equ3-12}
\int_{\mathbb{R}^N} \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}\to +\infty\quad
 \text{as }\ n\to\infty.
\end{equation}
It follows from \eqref{equ3-4} and \eqref{equ3-12} that
\begin{align*}
0&=\lim_{n\to \infty} \frac{c+o(1)}{\|v_{n}\|_{0}^{p}}
 = \lim_{n\to \infty} \frac{I(v_{n})}{\|v_{n}\|_{0}^{p}}\\
&=\lim_{n\to \infty} \frac{1}{\|v_{n}\|_{0}^{p}}
 \Big(\frac{1}{p} \int_{\mathbb{R}^N}(|\nabla v_{n}|^{p}
 + \overline{V}(x)|f(v_{n})|^{p}){\rm d}x
 - \int_{\mathbb{R}^N}\overline{G}(x,f(v_{n})){\rm d}x \Big)\\
&=\lim_{n\to \infty} \Big(\frac{1}{p}
 - \int_{\Omega_{n}(0,r_{0})}\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{p}}
 |\widetilde{f}(v_{n})|^{p}{\rm d}x \\
&\quad -   \int_{\Omega_{n}(r_{0}, +\infty)}
 \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{p}}
 |\widetilde{f}(v_{n})|^{p}{\rm d}x \Big)\\
&\leq \frac{1}{p}+ \limsup_{n\to \infty}(C_{22}
 + C_{23}r^{q-p}_{0})\int_{\mathbb{R}^N} |\widetilde{f}(v_{n})|^{p}{\rm d}x\\
&\quad -\int_{\Omega_{n}(r_{0}, +\infty)}
 \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{p}}|\widetilde{f}(v_{n})|^{p}{\rm d}x\\
&\leq C_{24}- \liminf_{n\to \infty} \int_{\mathbb{R}^N}
 \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}
 |f(v_{n}) \widetilde{f}(v_{n})|^{p}{\rm d}x
 = -\infty,
\end{align*}
which is a contradiction. Thus, there exists $C_{15}> 0$ such that
\[
\int_{\mathbb{R}^N}(|\nabla v_{n}|^{p}+ \overline{V}(x) |f(v_{n})|^{p}){\rm d}x
\leq C_{15}.
\]
Hence, from Proposition \ref{pro2-2}, we have that $\{v_{n}\}$ is bounded in $E$.
\end{proof}

\begin{lemma}\label{lem3-3}
Suppose that {\rm (A1'), (A2), (A3), (A5)} are satisfied. 
Then any $(C)_{c}$ sequence of $I$ is bounded.
\end{lemma}

\begin{proof}
Let $\{v_{n}\}\subset E$ be such that
\begin{equation}\label{equ3-13}
I(v_{n})\to c\quad \text{and}\quad  (1+ \|v_{n}\|_{E})I'(v_{n})\to 0.
\end{equation}
Thus, there is a constant $C_{25}> 0$ such that
\begin{equation}\label{equ3-14}
I(v_{n})-\frac{1}{\mu}\langle I'(v_{n}), v_{n}\rangle\leq C_{25}.
\end{equation}
Firstly, we prove that there exists $C_{26}> 0$ independent of $n$ such that
\[
\int_{\mathbb{R}^N}\Big(|\nabla v_{n}|^{p}
+ \overline{V}(x)|f(v_{n})|^{p}\Big){\rm d}x  \leq C_{26}.
\]
Suppose by contradiction, we assume that
\[
\|v_{n}\|^{p}_{0}:= \int_{\mathbb{R}^N}\Big(|\nabla v_{n}|^{p}
+ \overline{V}(x)|f(v_{n})|^{p}\Big){\rm d}x \to \infty\quad \text{as }
 n\to\infty.
\]
As
\[
\nabla (\frac{f(v_{n})}{f'(v_{n})})
= \nabla \Big[f(v_{n})\cdot(1+2^{p-1}|f(v_{n})|^{p})^{1/p}\Big]
=\nabla v_{n}[1+\frac{2^{p-1}|f(v_{n})|^{p}}{1+2^{p-1}|f(v_{n})|^{p}}].
\]
By (A5) and $\mu> 2p$ we can obtain
\begin{equation}\label{equ3-15}
\begin{split}
C_{25}
&\geq I(v_{n})- \frac{1}{\mu}\langle I'(v_{n}),\frac{f(v_{n})}{f'(v_{n})}\rangle\\
&= \frac{1}{p}\int_{\mathbb{R}^N}(|\nabla v_{n}|^{p}
 + \overline{V}(x)|f(v_{n})|^{p}){\rm d}x
 - \int_{\mathbb{R}^N}\overline{G}(x,f(v_{n})){\rm d}x\\
&\quad -\frac{1}{\mu}\int_{\mathbb{R}^N}\Big(|\nabla v_{n}|^{p-2}
 \nabla(v_{n})\nabla(\frac{f(v_{n})}{f'(v_{n})})\Big){\rm d}x\\
&\quad +\frac{1}{\mu}\int_{\mathbb{R}^N}
 \Big(\overline{g}(x,f(v_{n}))f'(v_{n})\frac{f(v_{n})}{f'(v_{n})}\Big){\rm d}x\\
&\quad -\frac{1}{\mu}\int_{\mathbb{R}^N}
 \Big( \overline{V}(x)|f(v_{n})|^{p-2}f(v_{n})f'(v_{n})\frac{f(v_{n})}{f'(v_{n})}
 \Big){\rm d}x\\
&=\int_{\mathbb{R}^N}\Big[\frac{1}{p}- \frac{1}{\mu}
 \Big(1+ \frac{2^{p-1}|f(v_{n})|^{p}}{1+2^{p-1}|f(v_{n})|^{p}}\Big)\Big]
 |\nabla v_{n}|^{p}{\rm d}x\\
&\quad + \int_{\mathbb{R}^N}(\frac{1}{p}
 - \frac{1}{\mu})(\overline{V}(x)|f(v_{n})|^{p}){\rm d}x \\
&\quad + \frac{1}{\mu}\int_{\mathbb{R}^N}\Big[\overline{g}(x,f(v_{n}))f(v_{n})
 - \mu \overline{G}(x,f(v_{n}))\Big]{\rm d}x \\
&\geq \int_{\mathbb{R}^N}(\frac{1}{p}- \frac{2}{\mu})|\nabla v_{n}|^{p}{\rm d}x
 + \int_{\mathbb{R}^N}(\frac{1}{p}
 - \frac{1}{\mu})(\overline{V}(x)|f(v_{n})|^{p}){\rm d}x \\
&\quad - \frac{1}{\mu}\int_{\mathbb{R}^N}|f(v_{n})|^{p}{\rm d}x\\
&\geq (\frac{1}{p}- \frac{2}{\mu})\int_{\mathbb{R}^N}
 \Big(|\nabla v_{n}|^{p}+ \overline{V}(x)|f(v_{n})|^{p}\Big){\rm d}x
 - \frac{1}{\mu}\int_{\mathbb{R}^N}|f(v_{n})|^{p}{\rm d}x\\
&\geq (\frac{1}{p}- \frac{2}{\mu})\|v_{n}\|_{0}^{p}
 - \frac{1}{\mu}\int_{\mathbb{R}^N}|f(v_{n})|^{p}{\rm d}x.
\end{split}
\end{equation}

Setting $\widetilde{f}(v_{n}):= f(v_{n})/\|v_{n}\|_{0}$, 
we have $\|\widetilde{f}(v_{n})\|_{E}\leq 1$. 
Passing to a subsequence, we may assume that 
$\widetilde{f}(v_{n})\rightharpoonup w$ in $E$, $\widetilde{f}(v_{n})\to w$ 
in $L^{s}(\mathbb{R}^N)$, $p \leq s < p^{\ast}$, and 
$\widetilde{f}(v_{n})\to w$ a.e. $\mathbb{R}^N$.

From \eqref{equ3-15},
\[
\frac{C_{25}}{\|v_{n}\|^{p}_{0}}\geq (\frac{1}{p}- \frac{2}{\mu})
- \frac{1}{\mu}\int_{\mathbb{R}^N}|\widetilde{f}(v_{n})|^{p}{\rm d}x.
\]
Hence, we obtain
\[
\frac{1}{\mu}\int_{\mathbb{R}^N}|\widetilde{f}(v_{n})|^{p}{\rm d}x
\geq (\frac{1}{p}- \frac{2}{\mu})\mu + o(1).
\]
Then $\widetilde{f}(v_{n})\to w$ and $w\neq 0$, so
$|f(v_{n})|\to \infty$ as $n\to \infty$.

Also by (A3), we have
\[
\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}\to +\infty.
\]
So
\begin{equation}\label{equ3-16}
\int_{\mathbb{R}^N}\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}\to +\infty,
\end{equation}
From \eqref{equ3-13} and \eqref{equ3-16} it follows that
\begin{equation}\label{equ3-17}
\begin{split}
0&=\lim_{n\to \infty} \frac{c+o(1)}{\|v_{n}\|_{0}^{p}}
 = \lim_{n\to \infty} \frac{I(v_{n})}{\|v_{n}\|_{0}^{p}}\\
&=\lim_{n\to \infty} \frac{1}{\|v_{n}\|_{0}^{p}}
 \Big(\frac{1}{p} \int_{\mathbb{R}^N}(|\nabla v_{n}|^{p}
 + \overline{V}(x)|f(v_{n})|^{p}){\rm d}x
 - \int_{\mathbb{R}^N}\overline{G}(x,f(v_{n})){\rm d}x \Big)\\
&=\lim_{n\to \infty}\Big(\frac{1}{p}
 - \int_{\mathbb{R}^N}\frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{p}}
 |\widetilde{f}(v_{n})|^{p}{\rm d}x \Big)\\
&\leq C_{27}- \liminf_{n\to \infty}
 \int_{\mathbb{R}^N} \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}}
 |f(v_{n}) \widetilde{f}(v_{n})|^{p}{\rm d}x
= -\infty.
\end{split}
\end{equation}
Which is a contradiction. Thus, there exists $C_{26}> 0$ such that
\[
\int_{\mathbb{R}^N}(|\nabla v_{n}|^{p}+ \overline{V}(x)
|f(v_{n})|^{p}){\rm d}x \leq C_{26}.
\]
Hence, from Proposition \ref{pro2-2}, we obtain that $\{v_{n}\}$ is bounded in $E$.
\end{proof}

Since (A2) and (A6) imply  $(A5)$, we have the following corollary.

\begin{corollary}\label{cor3-1}
Suppose that {\rm (A1'), (A2), (A3), (A6)} are satisfied. 
Then any $(C)_{c}$ sequence of $I$ is bounded.
\end{corollary}

\section{Proof of main results}\label{3}

\subsection*{Proof of Theorems \ref{thm1-1} and \ref{thm1-2}}

\begin{lemma}\label{lem4-1}
The functional $I$ is bounded from below on a neighborhood of the origin. That is,
there exist $C_{28}\in \mathbb{R}$ and $\rho> 0$, such that
\[
I(u)\geq C_{28},\quad \forall u\in B_{\rho}= \{u\in E:\|u\|\leq \rho\}.
\]
\end{lemma}

\begin{proof}
If the conclusion is not true, there exists $\{u_{n}\}\subset E$, satisfying
\[
\|u_{n}\|\leq \frac{1}{n},\quad  I(u_{n})\to -\infty.
\]
So $u_{n}\to 0$ in $E$, and
\[
I(u_{n})= \frac{1}{p}\int_{\mathbb{R}^N}(|\nabla u_{n}|^{p}
+ \overline{V}(x)|f(u_{n})|^{p}){\rm d}x
- \int_{\mathbb{R}^N}\overline{G}(x,f(u_{n})){\rm d}x.
\]
Obviously,
\[
\frac{1}{p}\int_{\mathbb{R}^N}(|\nabla u_{n}|^{p}
+ \overline{V}(x)|f(u_{n})|^{p}){\rm d}x\to 0.
\]
From (A2), and (3) and (8) of Lemma \ref{lem2-2}, we have
\begin{align*}
\int_{\mathbb{R}^N}\overline{G}(x,f(u_{n})){\rm d}x
&\leq C_{29}\int_{\mathbb{R}^N}\big(|f(u_{n})|^{p}+ |f(u_{n})|^{q}\big){\rm d}x\\
&\leq C_{29}\int_{\mathbb{R}^N}(|u_{n}|^{p}+ |u_{n}|^{\frac{q}{2}}){\rm d}x\to 0.
\end{align*}
Hence, $I(u_{n})\to 0$, contradicts with $I(u_{n})\to -\infty$,
as $n\to +\infty$.
\end{proof}

\begin{lemma}\label{lem4-2}
There exists $\vartheta\in E$, such that $I(t\vartheta)< 0$, for $t$ small enough.
\end{lemma}

\begin{proof}
Let $\vartheta\in C^{\infty}_{0}(\mathbb{R}^N, [0,1])\backslash\{0\}$, and 
$K= \operatorname{supp} \vartheta$. From (A3), we have
\[
\overline{G}(x,u)\geq C_{30}|u|^{\tau}> 0,
\]
for all $(x,u)\in\mathbb{R}^N\times\mathbb{R}$, $|u|\geq r_0$.
By $(A2)$, for a.e. $x\in\mathbb{R}^N$ and $0\leq |u|\leq 1$,
there exists $M> 0$ such that
\[
|\frac{\overline{g}(x,u)u}{|u|^{p}}|
\leq \big|\frac{C( |u|^{p-1} + |u|^{q-1})\cdot|u|}{|u|^{p}}\big|\leq M,
\]
which implies that
\[
\overline{g}(x,u)u\geq -M |u|^{p}.
\]
We can use the equality $\overline{G}(x,u)=\int_0^1\overline{g}(x,tu)u{\rm d}t$,
for a.e. $x\in\mathbb{R}^N$ and $0\leq |u|\leq 1$, to obtain
\[
\overline{G}(x,u)\geq -\frac{M}{p}|u|^{p}\,.
\]
Then
\begin{equation}\label{equ4-1}
\overline{G}(x,u)\geq -\frac{M}{p}|u|^{p}+ C_{30}|u|^{\tau}.
\end{equation}
So from  \eqref{equ4-1},
\begin{equation}\label{equ4-2}
\begin{split}
&I(t\vartheta)\\
&= \frac{t^{p}}{p}\int_{\mathbb{R}^N}|\nabla \vartheta|^{p}{\rm d}x
 + \frac{1}{p}\int_{\mathbb{R}^N}\overline{V}(x)|f(t\vartheta)|^{p}{\rm d}x
 - \int_{\mathbb{R}^N}\overline{G}(x,f(t\vartheta)){\rm d}x\\
&\leq \frac{t^{p}}{p}\int_{\mathbb{R}^N}\big(|\nabla \vartheta|^{p}
 + \overline{V}(x)|\vartheta|^{p}\big){\rm d}x+ \frac{M}{p}\int_{\mathbb{R}^N}|f(t\vartheta)|^{p}{\rm d}x- C_{31} \int_{\mathbb{R}^N}|f(t\vartheta)|^{\tau}{\rm d}x\\
&\leq \frac{t^{p}}{p}\int_{\mathbb{R}^N}\big(|\nabla \vartheta|^{p}
 + \overline{V}(x)|\vartheta|^{p}+ M |\vartheta|^{p}\big){\rm d}x
 - C_{31} \int_{\mathbb{R}^N}|f(t\vartheta)|^{\tau}{\rm d}x.
\end{split}
\end{equation}

Since $f(t)/t$ is decreasing and $0\leq t\vartheta \leq t$, for $t\geq 0$. 
We obtain $f(t\vartheta)\geq f(t)\vartheta$. By (9) of Lemma \ref{lem2-2}, 
we obtain $f(t\vartheta)\geq Ct\vartheta$, for $0\leq t\leq 1$.
Hence
\[
I(t\vartheta)\leq \frac{t^{p}}{p}
\int_{\mathbb{R}^N}\big(|\nabla \vartheta|^{p}
+ \overline{V}(x)|\vartheta|^{p}+ M |\vartheta|^{p}\big){\rm d}x
- C_{32}t^{\tau} \int_{\mathbb{R}^N}|\vartheta|^{\tau}{\rm d}x,
\]
and since $\tau< p$, we obtain $I(t\vartheta)< 0$, for $t$ sufficiently  small
and the Lemma is proved.
\end{proof}

Thus, we obtain that
\[
c_{0}= \inf\{I(u): u\in \overline{B_{\rho}}\}< 0,
\]
which $\rho> 0$ is given in Lemma \ref{lem4-1}.
Then we can apply the Ekeland's variational principle and
\cite[corollary 2.5]{Willem}, there exists a sequence
$\{u_{n}\}\subset \overline{B_{\rho}}$ such that
$C_{33}\leq I(u_{n})< C_{33}+ \frac{1}{n}$.
Hence
\[
I(u)\geq I(u_{n})- \frac{1}{n}\|w- u_{n}\|_{E},\quad \forall
 w\in \overline{B_{\rho}}.
\]
Then, following the idea in \cite{Willem}, we can show that $\{u_n\}$
is a bounded Cerami sequence of $I$. Therefore,
Lemma \ref{lem3-1} implies that there exists a function
$u_0\in E$ such that $I'(u_0)=0$ and $I(u_0)=c_0 < 0$.

Next, we show that there exists a second solution for problem \ref{equ1-1}.

\begin{lemma}\label{lem4-3}
If the conditions {\rm (A1)--(A3), (A7)} are satisfied, there exist two 
constants $\rho_1> 0$, $\alpha> 0$, such that
\[
I(u)\geq\alpha> 0,\quad \forall  u\in S_{\rho_1}=\{u\in E: \|u\|_{E}=\rho_1\}.
\]
\end{lemma}

\begin{proof}
From (A2) and (A7), it follows that
\[
|\overline{G}(x,u)|\leq\varepsilon|u|^p+C_{\varepsilon}|u|^q,\quad
\forall (x,u)\in\mathbb{R}^N\times\mathbb{R}.
\]
Thus, by Proposition \ref{pro2-1}, we take $u\in E$ with $\|u\|\leq \rho$,
 where $\rho$ is given in Proposition \ref{pro2-1}, we can deduce that
\begin{equation}\label{equ4-3}
\begin{split}
I(u)&= \frac{1}{p}\int_{\mathbb{R}^N}(|\nabla u|^{p}
 + \overline{V}(x)|f(u)|^{p}){\rm d}x
 - \int_{\mathbb{R}^N}\overline{G}(x,f(u)){\rm d}x\\
&\geq \frac{C_{34}}{p}\|u\|_E^{p}- C\varepsilon\|u\|^{p}_{E}
 - C_{\varepsilon}\|u\|_E^{q}\\
&\geq\frac{C_{35}}{2p}\|u\|_E^p-C_{36}\|u\|_E^q,
\end{split}
\end{equation}
and since $q> 2p$, there exists $\alpha,\rho_1>0$ such that
$I(u)\geq \alpha> 0$ for $\|u\|_E=\rho_1$.
\end{proof}

\begin{lemma}\label{lem4-4}
There exist a $v\in E$ with $\|v\|_{E}>\rho_1$, such that $I(v)< 0$, 
which $\rho_{1}$ is defined in Lemma \ref{lem4-3}.
\end{lemma}

\begin{proof}
Let $u_0\in E$ and $u_0>0$. From $(A3)$, (9) of Lemma \ref{lem2-2}, and 
Fatou's Lemma, we have
\begin{align*}
\lim_{t\to \infty} \frac{I(tu_{0})}{t^{p}}
&= \lim_{t\to \infty}\Big(\frac{1}{pt^{p}} \int_{\mathbb{R}^N}(|\nabla tu_{0}|^{p} 
 + \overline{V}(x)|f(tu_{0})|^{p}){\rm d}x  
 - \int_{\mathbb{R}^N}\frac{\overline{G}(x,f(tu_{0}))}{t^{p}}{\rm d}x\Big)\\
&\leq \lim_{t\to \infty}\Big(\int_{\mathbb{R}^N}\frac{|\nabla u_{0}|^{p}}{p}{\rm d}x
 + \int_{\mathbb{R}^N}\frac{\overline{V}(x)|tu_{0}|^{p}}{pt^{p}}{\rm d}x\\
&- \int_{\mathbb{R}^N}\frac{\overline{G}(x,f(tu_{0}))}{(f(tu_{0}))^{2p}} 
 \frac{(f(tu_{0}))^{2p}}{(tu_{0})^{p}}(u_{0})^{p}{\rm d}x\Big)\\
&=\frac{\|u_{0}\|^{p}_{E}}{p}-\lim_{t\to \infty}\int_{\mathbb{R}^N}
 \frac{\overline{G}(x,f(tu_{0}))}{(f(tu_{0}))^{2p}} 
 \frac{(f(tu_{0}))^{2p}}{(tu_{0})^{p}}(u_{0})^{p}{\rm d}x\\
&\leq\frac{\|u_{0}\|^{p}_{E}}{p}-\int_{\mathbb{R}^N}
 \liminf_{t\to \infty}\frac{\overline{G}(x,f(tu_{0}))}{(f(tu_{0}))^{2p}}
  \frac{(f(tu_{0}))^{2p}}{(tu_{0})^{p}}(u_{0})^{p}{\rm d}x
= -\infty.
\end{align*}
Thus, this lemma is proved by taking $v= tu_{0}$ with $t> 0$ large enough.
\end{proof}

Based on Lemmas \ref{lem4-3} and \ref{lem4-4}, Theorem \ref{thm2-1} 
implies that there is a sequence $\{u_{n}\}\subset E$ such that
\[
I(u_{n})\to c\quad \text{and}\quad (1+ \|u_{n}\|_{E})I'(u_{n})\to 0.
\]
From Lemma \ref{lem3-2} and \ref{lem3-1}, it shows that this sequence
$\{u_{n}\}$ has a convergent subsequence in $E$. Thus, there exists
$u_{1}\in E$ such that $I'(u_{1})=0$ and $I(u_{1})=c_{1}>0$.
Consequently, the proof of Theorem \ref{thm1-1} is complete.

By the similar arguments as the proof of Theorem \ref{thm1-1}, 
Theorem \ref{thm1-2} and Corollary \ref{cor1-1} can be proved.

\subsection*{Proof of Theorems \ref{thm1-3} and \ref{thm1-4}}


Let $\{e_{i}\}_{i\in  \mathbb{N}}\in E$ is a total orthonormal basis of $E$ 
and $\{e^{\ast}_{j}\}_{j\in  \mathbb{N}}\in E^{\ast}$, so that
\begin{gather*}
E= \overline{\operatorname{span}\{ e_{i}: i=1,2,\cdots\}},\quad
   E^{\ast}= \overline{\operatorname{span}\{ e^{\ast}_{j}: j=1,2,\cdots\}}, \\
\langle e_{i}, e^{\ast}_{j}\rangle =
\begin{cases}
1,& i=j,\\
0,& i\neq j;
\end{cases}
\end{gather*}
So we define $X_{j}= \mathbb{R}e_{j}$,
\[
Y_{k}= \oplus^{k}_{j=1} X_{j},\quad
Z_{k}=  \overline{\oplus^{\infty}_{j=k+1} X_{j}} ,\quad  k\in \mathbb{Z}
\]
and $Y_{k}$ is finite-dimensional. Similar to \cite[Lemma 3.8]{Willem},
 we have the following lemma.

\begin{lemma}\label{lem4-5}
Under assumption {\rm (A1')}, for $p\leq s< p^{\ast}$,
\[
\beta_{k}(s):= \sup_{v\in Z_{k},\|v\|=1}  \|v\|_{s} \to 0,\quad  k\to \infty .
\]
\end{lemma}

\begin{lemma}\label{lem4-6}
Suppose that {\rm (A1'), (A2)} are satisfied. Then there exist constants 
$\rho> 0$, $\alpha> 0$ such that $I\big|_{ S_{\rho}\bigcap Z_{m}}\geq \alpha $.
\end{lemma}

\begin{proof}
For any $v\in Z_{m}$ with $\|v\|_{E}=\rho < 1$, by 
(3) and (8) of Lemma \ref{lem2-2}, and proposition \ref{pro2-1}, we have
\begin{equation}\label{equ4-4}
\begin{split}
I(v)
&= \frac{1}{p} \int_{\mathbb{R}^N} (|\nabla v|^{p}
+  \overline{V}(x)|f(v)|^{p}){\rm d}x 
- \int_{\mathbb{R}^N} \overline{G}(x,f(v)){\rm d}x \\
&\geq  \frac{C_{37}}{p} \|v\|^{p}_{E}
 -C_{38} \int_{\mathbb{R}^N} (|f(v)|^{p} + |f(v)|^{q}) {\rm d}x \\
&\geq \frac{C_{37}}{p} \|v\|^{p}_{E}
 - C_{39}\int_{\mathbb{R}^N}(|v|^{p} + |v|^{\frac{q}{2}}){\rm d}x.
\end{split}
\end{equation}
By Lemma \ref{lem4-5}, we can choose an integer $m\geq 1$ such that
\[
C_{39}\|v\|^{p}_{p}\leq \frac{C_{37}}{2p}\|v\|^{p}_{E},\quad
C_{39}\|v\|^{\frac{q}{2}}_{\frac{q}{2}}\leq \frac{C_{37}}{2p}\|v\|^{\frac{q}{2}}_{E},
\quad \forall v \in Z_{m}.
\]
Combining the above inequality with \eqref{equ4-4}, we have
\[
I(v) \geq \frac{C_{37}}{p} \|v\|^{p}_{E}- \frac{C_{37}}{2p}\|v\|^{p}_{E}
 - \frac{C_{37}}{2p}\|v\|^{\frac{q}{2}}_{E}
= \frac{C_{37}}{2p} \|v\|^{p}_{E}(1- \|v\|^{\frac{q-2p}{2}}_{E})> 0,
\]
since $q>2p$. This completes the proof.
\end{proof}


\begin{lemma}\label{lem4-7}
Suppose that {\rm (A1'), (A2), (A3)} are satisfied. Then
for any finite dimensional subspace $\widetilde{E}\subset E$, 
there is $R= R(\widetilde{E})> 0$ such that
\[
I(v)\leq 0,\quad   \forall v\in \widetilde{E}\backslash B_{R}.
\]
\end{lemma}

\begin{proof}
For any finite dimensional subspace $\widetilde{E}\subset E$, there 
exists a $m\in\mathbb{N}$ such that $\widetilde{E}\subset E_{m}$. 
Suppose by contradiction, we assume that there exists a sequence 
$\{v_{n}\}\subset \widetilde{E}$ such that $\|v_{n}\|_{E}\to \infty$ and 
$I(v_{n})> 0$. Hence
\begin{equation}\label{equ4-5}
\frac{1}{p} \int_{\mathbb{R}^N} (|\nabla v_{n}|^{p}
+  \overline{V}(x)|f(v_{n})|^{p}){\rm d}x 
>  \int_{\mathbb{R}^N}\overline{ G}(x,f(v_{n})){\rm d}x.
\end{equation}
Set $w_{n}= \frac{v_{n}}{\|v_{n}\|_{E}}$. Then, up to a subsequence, 
we can assume that $w_{n}\rightharpoonup w$ in $E$, $w_{n}\to w$ 
in $L^{s}(\mathbb{R}^N)$ for all $p\leq s< p^{\ast}$, and 
$w_{n}\to w$ a.e.on $\mathbb{R}^N$. 
Set $\Omega_{1}:= \{x\in \mathbb{R}^N: w(x) \neq 0 \}$ 
and $\Omega_{2}:= \{x\in \mathbb{R}^N: w(x) = 0 \}$. 
If $\operatorname{meas}(\Omega_{1})> 0$, by (A3), (5)
of Lemma \ref{lem2-2},  and Fatou's lemma, we have
\begin{equation}\label{equ4-6}
\int_{\Omega_{1}} \frac{\overline{G}(x,f(v_{n}))}{\|v_{n}\|^{p}_{E}}{\rm d}x 
= \int_{\Omega_{1}} \frac{\overline{G}(x,f(v_{n}))}{|f(v_{n})|^{2p}} 
\frac{|f(v_{n})|^{2p}}{|v_{n}|^{p}}|w_{n}|^{p} {\rm d}x \to +\infty.
\end{equation}
On the other hand, by (A2) and (A3), there exists $C_{40}>0$ such that
\[
\overline{G}(x,t)\geq- C_{40}|t|^p,\quad \text{for all }
 (x,t)\in\mathbb{R}^N\times\mathbb{R}.
\]
Hence
\[
\int_{\Omega_{2}} \frac{\overline{G}(x,f(v_{n}))}{\|v_{n}\|^{p}_{E}}{\rm d}x
\geq -C_{40}\int_{\Omega_{2}}\frac{|f(v_{n})|^p}{\|v_{n}\|^{p}_{E}}{\rm d}x
\geq-C_{41}\int_{\Omega_{2}}|w_n|^p {\rm d}x.
\]
Hence, by the fact that $w_{n}\to w$ in $L^{p}(\mathbb{R}^N)$, we obtain
\[
\liminf_{n \to \infty} \int_{\Omega_{2}}
\frac{\overline{G}(x,f(v_{n}))}{\|v_{n}\|^{p}_{E}}{\rm d}x \geq 0.
\]
Combining this with \eqref{equ4-6}, we have
\[
\int_{\mathbb{R}^N} \frac{\overline{G}(x,f(v_{n}))}{\|v_{n}\|^{p}_{E}}{\rm d}x
= +\infty,
\]
which implies a contradiction with \eqref{equ4-5}.
Hence, $\operatorname{meas}(\Omega_{1})= 0$, i.e. $w(x)= 0$
a.e. on $\mathbb{R}^N$. By the fact that all norms are equivalent in
$\widetilde{E}$, there exists $C_{42}> 0$ such that
\[
\|v\|^{p}_{p} \geq C_{42}\|v\|^{p}_{E},\quad  \forall v\in \widetilde{E}.
\]
Hence
\[
0= \lim_{n \to \infty} \|w_{n}\|^{p}_{p}
\geq \lim_{n \to \infty} C_{42}\|w_{n}\|^{p}_{E}=C_{42},
\]
this results in a contradiction. The proof is complete.
\end{proof}


\begin{proof}[Proof of theorem \ref{thm1-1}]
Let $X= E$, $Y= Y_{m}$ and $Z= Z_{m}$. Obviously, $I(0)= 0$ and 
(A8) imply that $I$ is even. 
By Lemma \ref{lem3-2}, Lemma \ref{lem4-2} and Lemma \ref{lem4-3}, 
all conditions of Theorem \ref{thm2-2} are satisfied. 
Thus, problem \eqref{equ2-1} possesses infinitely many nontrivial solutions 
$\{v_{n}\}$ such that $I(v_{n})\to \infty$ as $n\to \infty$. 
Namely, problem \eqref{equ1-1} also possesses infinitely many nontrivial 
solutions  $\{u_{n}\}$ such that $J(u_{n})\to \infty$ as $n\to \infty$.
\end{proof}

By the similar arguments as Theorem \ref{thm1-3}, we can give 
the proof of Theorem \ref{thm1-4} and Corollary \ref{cor1-2}.

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation
of China (NSFC 11501403), and by the Shanxi Province Science Foundation
for Youths under grant 2013021001-3.

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