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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 05, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/05\hfil Existence of nontrivial solutions]
{Existence of nontrivial solutions for a quasilinear Schr\"odinger
equations with sign-changing potential}

\author[X.-D. Fang, Z.-Q. Han \hfil EJDE-2014/05\hfilneg]
{Xiang-Dong Fang, Zhi-Qing Han}  % in alphabetical order

\address{Xiang-Dong Fang \newline
School of Mathematical Sciences, Dalian University of Technology\\
116024 Dalian,  China}
\email{fangxd0401@gmail.com, Phone +86 15840980504}

\address{Zhi-Qing Han (Corresponding author)\newline
School of Mathematical Sciences, Dalian University of Technology\\
116024 Dalian,  China}
\email{hanzhiq@dlut.edu.cn}

\thanks{Submitted September 13, 2013. Published January 3, 2014.}
\subjclass[2000]{35A01, 35A15, 35Q55}
\keywords{Quasilinear Schr\"odinger equation; sign-changing potential;
 \hfill\break\indent Cerami sequences}

\begin{abstract}
 In this article we consider the  quasilinear Schr\"odinger
 equation where the potential is sign-changing. We employ a mountain
 pass argument without compactness conditions to obtain the existence
 of a nontrivial solution.
\end{abstract}

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

\section{Introduction}

In this paper we are concerned with the existence of a nontrivial solution 
for the quasilinear Schr\"odinger equation
\begin{equation} \label{f1}
-\Delta u+V(x)u-\Delta(u^{2})u=f(x,u), \quad x\in \mathbb{R}^{N}
\end{equation}  
These type of equations come from the study of the standing wave
solutions of quasilinear Schr\"odinger equations derived as models
for several physical phenomena; see \cite{poppenberg}.
The case $\inf_{\mathbb{R}^N}V(x)>0$ has been extensively studied
in recent years. However, to our best knowledge, there is no result
for the other important  case $\inf_{\mathbb{R}^N}V(x)<0$.
The various methods developed for the quasilinear Schr\"{o}dinger equations
do not seem to apply directly in this case.

In this article, we assume that the potential is sign-changing 
and the nonlinearity is more general than in other articles. 
Some authors recover the compactness by assuming that
the potential $V(x)$ is either coercive or has radial symmetry,
 see \cite{deng,wang1,wang3,poppenberg}. 
Here we do not need the compactness, but we assume the potential 
bounded from above, but may be unbounded from below. 
We consider the case $\mathbb{N}\geq 3$.
This work is motivated by the ideas in \cite{colin1,silva,szulkin,zhang}.


First we consider the problem
\begin{equation}
\label{f1b}
-\Delta u+V(x)u-\Delta(u^{2})u=g(x,u)+h(x), \quad u\in H^1(\mathbb{R}^N).
\end{equation}
We suppose that $V$ and $g$ satisfy the following assumptions:
\begin{itemize}
\item[(G1)] $g$ is continuous, and $|g(x,u)|\leq a(1+|u|^{p-1})$ for some
$a>0$ and $4<p<2\cdot 2^*$, where $2^*:=2N/(N-2)$.
\item[(G2)] $g(x,u)=o(u)$ uniformly in $x$ as $u\to 0$.
\item[(G3)] There exists $\theta >4$ such that $0< \theta G(x,u)\leq g(x,u)u$,
 $\text{for} \ x\in \mathbb{R}^N$, $u\in \mathbb{R}\backslash \{0\}$,
 where $G(x,u):=\int_{0}^{u}g(x,s)ds$.
\item[(V1)] $V(x)$ is sign-changing, $V^+(x)\in L^{\infty}(\mathbb{R}^N)$,
 $\lim_{|x|\to\infty}V^+(x)=a_0>0$ and
 $|V^{-}|_{L^{N/2}(\mathbb{R}^N)}<\frac{\theta -4}{S(\theta -2)}$,
 where $V^{\pm}(x):=\max\{\pm V(x),0\}$, and $S$ denotes the Sobolev optimal
 constant.
\item[(V2)] $\int_{\mathbb{R}^N}|\nabla u|^2+V(x)u^2>0$ for every
 $u\in E\setminus \{0\}$.
\item[(H1)] $h\neq 0$ and $|h|_{L^{2N/(N+2)}}<S^{-1/2}k\rho$, where $k$ and
 $\rho$ are given in Lemma \ref{lem3.3}.
\end{itemize}
We remark that (V2) means that either $V^-$ is small or $V^-$ has a small
support. It is also easy to give a concrete condition on $V^-$ satisfying
(V2).
Note that (H1) is similar to  condition (H8) in \cite{zhang}.
Let
$$
E:=\big\{u\in H^1(\mathbb{R}^N):\int_{\mathbb{R}^N}V^{+}(x)u^{2}dx
<\infty\big\},
$$
with norm $\|u\|:=\big(\int_{\mathbb{R}^N}|\nabla u|^{2}+V^+(x)u^2\big)^{1/2}$.
It is obvious that $\|\cdot\|$ is an equivalent norm with the standard one
by (V1). In Section 3 we prove one of  the main result of this article:

\begin{theorem} \label{thm1.1}
Suppose that {\rm (V1), (V2), (G1)--(G3), (H1)} are satisfied. 
Then \eqref{f1} has a nontrivial solution.
\end{theorem}

\begin{remark} \label{rmk1} \rm
 Under stronger conditions on $V$,
 we can  derive the existence of a nontrivial solution without 
the small perturbation term $h$. 
For example, if  $V$ is continuous, 
1-periodic in $x_i$, $1\leq i\leq N$, and there exists a constant $a_0>0$ 
such that $V(x)\geq a_0$ for all $x\in \mathbb{R}^N$. 
It is obvious that the condition in Theorem \ref{thm1.1} is satisfied,
so we have a bounded $(C)_c$ sequence by the proof of Theorem \ref{thm1.1}. 
Similarly as in \cite[Lemma 1.2]{silva}, under a translation 
if necessary, we get a nontrivial solution.
\end{remark}

Also, we consider the  problem
\begin{equation} \label{p2}
-\Delta u+V(x)u-\Delta(u^{2})u=g(u), \quad u\in H^1(\mathbb{R}^{N}),
\end{equation}
where the nonlinearity $g$ satisfies (G1)--(G3). We assume that
\begin{itemize}
\item[(V1')] $V(x)$ is sign-changing,
$\lim_{|x|\to\infty}V^+(x)=V^+(\infty)>0$, $V^+(x)\leq V^+(\infty)$ on
$\mathbb{R}^N$ and
$|V^{-}|_{L^{N/2}(\mathbb{R}^N)}<(\theta -4)/(S(\theta -2))$.
\end{itemize}
Note that (V1') implies (V1).
The second main result of this paper, which we prove in Section 4,
is the following.

\begin{theorem} \label{thm1.2} 
Suppose that {\rm (V1'), (V2)} are satisfied. Then
\eqref{p2} admits a nontrivial solution.
\end{theorem}


\begin{remark} \label{rmk2} \rm
We would like to point out that (V1') is weaker than the assumptions 
(V0) and (V1)  in \cite{colin1}. 
But they obtain the existence of a positive solution, while we do not.
\end{remark}

Positive constants will be denoted by  $C, C_{1}, C_{2},\dots$, while 
$|A|$  will denote the Lebesgue measure of a  set $A\subset\mathbb{R}^N$.

\section{Preliminary results} \label{pr}

We observe that  \eqref{f1} is the Euler-Lagrange equation associated
 with the energy functional
\begin{equation}
\label{f2}
J(u) :=\frac{1}{2}\int_{\mathbb{R}^N}(1+2u^{2})|\nabla u|^{2}
+\frac{1}{2}\int_{\mathbb{R}^N}V(x)u^{2}-\int_{\mathbb{R}^N}(G(x,u)+h(x)u).
\end{equation}
To use the usual argument, we make a change of variables $v:=f^{-1}(u)$,
 where $f$ is defined by
\[
f'(t)=\frac{1}{(1+2f^{2}(t))^{1/2}}\text{on } [0,+\infty) \quad
\text{and} \quad  f(t)=-f(-t) \text{on } (-\infty ,0].
\]
Below we summarize the properties of $f$, whose can be found
in \cite{colin1, severo, severo1}.

\begin{lemma} \label{lem2.1} 
The function $f$ satisfies the following properties:
\begin{itemize}
  \item[(1)] $f$ is uniquely defined, $C^{\infty}$ and invertible;
  \item[(2)] $|f'(t)|\leq 1$ for all $t\in \mathbb{R}$;
  \item[(3)] $|f(t)|\leq |t|$ for all $t\in \mathbb{R}$;
  \item[(4)] $f(t)/t\to 1$ as $t\to 0$;
  \item[(5)] $f(t)/\sqrt{t}\to 2^{1/4}$ as $t\to +\infty$;
  \item[(6)] $f(t)/2\leq tf'(t)\leq f(t)$ for all $t\in \mathbb{R}$;
  \item[(7)] $|f(t)|\leq 2^{1/4}|t|^{1/2}$ for all $t\in \mathbb{R}$;
  \item[(8)] $f^{2}(t)-f(t)f'(t)t\geq 0$ for all $t\in \mathbb{R}$;
  \item[(9)] there exists a positive constant $C$ such that 
 $|f(t)|\geq C|t|$ for $|t|\leq 1$ and $|f(t)|\geq C|t|^{1/2}$ for $|t|\geq 1$;
  \item[(10)] $|f(t)f'(t)| < 1/\sqrt{2}$ for all $t\in \mathbb{R}$.
\end{itemize}
\end{lemma}

Consider the functional
$$
I(v): =\frac{1}{2}\int_{\mathbb{R}^N}|\nabla v|^{2}
 +\frac{1}{2}\int_{\mathbb{R}^N}V(x)f^{2}(v)
 -\int_{\mathbb{R}^N}(G(x,f(v))+h(x)f(v)).
$$
Then $I$ is well-defined on $E$ and $I\in C^{1}(E,\mathbb{R})$ under the hypotheses
(V1), (G1) and (G2). It is easy to see that
\[ %\label{f4}
\langle I'(v),w\rangle
=\int_{\mathbb{R}^N} \nabla v \nabla w+\int_{\mathbb{R}^N}V(x)f(v)f'(v)w 
 -\int_{\mathbb{R}^N}(g(x,f(v))+h(x))f'(v)w
\]
for all $v,w\in E$ and
the critical points of $I$ are  weak solutions of the problem
$$
-\Delta v+V(x)f(v)f'(v)=(g(x,f(v))+h(x))f'(v), \quad v\in E.
$$
If $v\in E$ is a critical point of the functional $I$, then  
$u=f(v)\in E$ and $u$ is a solution of \eqref{f1}
 (cf: \cite{colin1}).

\section{Proof of Theorem \ref{thm1.1}} %\label{t1}

In the following we assume that (V1), (V2), (G1)--(G3) and (H1) are satisfied. 
First, (G1) and (G2) imply that for each $\varepsilon>0$ there is 
$C_\varepsilon >0$ such that
\begin{equation} \label{subcritical}
|g(x,u)|\leq \varepsilon |u|+C_{\varepsilon}|u|^{p-1}\quad \text{for all }
 u\in \mathbb{R}.
\end{equation}


\begin{lemma} \label{lem3.2} 
There exist $\xi,\alpha>0$ such that
 $\int_{\mathbb{R}^N}|\nabla u|^{2}+\int_{\mathbb{R}^N}V(x)f^2(u)
\geq \alpha \|u\|^2$, if $\|u\|=\xi$.
\end{lemma}

\begin{proof}
Arguing by contradiction, there exist $u_n\to 0$ in $E$, such that 
$$
\int_{\mathbb{R}^N}|\nabla v_{n}|^{2}+V(x)\frac{f^{2}(u_n)}{u^2_n}v^2_n\to 0.
$$
where $v_n:=\frac{u_n}{\|u_n\|}$.
We have that $u_n\to 0$ in $L^2(\mathbb{R}^N)$, $u_n\to 0$ a.e.,
 $v_n\rightharpoonup v$ in $E$, $v_n\to v$ in $L^{2}_{loc}$, 
$v_n\to v$ a.e. up to a subsequence.

If $v\neq 0$, then we claim that
$$
\liminf_{n\to \infty}\int_{\mathbb{R}^N}|\nabla v_{n}|^{2}
+V(x)\frac{f^{2}(u_n)}{u^2_n}v^2_n
\geq \int_{\mathbb{R}^N}|\nabla v|^{2}+V(x)v^2.
$$
Indeed, we have 
$$
\liminf_{n\to \infty}\int_{\mathbb{R}^N}V^+(x)\frac{f^{2}(u_n)}{u^2_n}v^2_n
\geq \int_{\mathbb{R}^N}V^+(x)v^2
$$ 
due to Fatou's lemma and Lemma \ref{lem2.1}-(4).
Since $v^2_n\rightharpoonup v^2$ in $L^{N/(N-2)}$ and $V^-(x)\in L^{N/2}$, 
we obtain 
$$
\int_{\mathbb{R}^N}V^-(x)\frac{f^{2}(u_n)}{u^2_n}v^2_n
\leq \int_{\mathbb{R}^N}V^-(x)v^2_n\to \int_{\mathbb{R}^N}V^-(x)v^2
$$
by Lemma \ref{lem2.1}(3) and the definition of the weak convergence. 
We have a contradiction to (V2).

The other case is $v=0$. Note that 
$\lim_{n\to \infty}\int_{\mathbb{R}^N}V^-(x)\frac{f^{2}(u_n)}{u^2_n}v^2_n=0$, 
then
$$
\int_{\mathbb{R}^{N}}(|\nabla v_{n}|^{2}+V^+(x)v_{n}^{2})
+\int_{\mathbb{R}^{N}}V^+(x)\Big(\frac{f^{2}(u_{n})}{u_{n}^{2}}-1\Big)
v_{n}^{2}\to 0.
$$
We use a similar argument as in \cite[Lemma 3.3]{fang}. 
Since $u_n\to 0$ in $L^2(\mathbb{R}^N)$, for every $\varepsilon >0$,
 $|\{x\in \mathbb{R}^N:|u_n(x)|>\varepsilon\}|\to 0$ as $n\to\infty$.
 We have by (V1), Lemma \ref{lem2.1}(3) and the H\"older inequality,
\begin{align*}
\big|\int_{|u_n|>\varepsilon}V^+(x)
\Big(\frac{f^{2}(u_{n})}{u_{n}^{2}}-1\Big)v_{n}^{2}\big|
&\leq C\int_{|u_n|>\varepsilon}v^2_n\\
&\leq |\{x\in \mathbb{R}^N:|u_n(x)|>\varepsilon\}|^{2/N}|v_n|^2_{2^*}\to 0.
\end{align*}
Now it follows from Lemma \ref{lem2.1}(4) and 
$\int_{\mathbb{R}^N}V^+(x)v^2_n\leq C_1$ that
$$
\int_{|u_n|<\varepsilon}V^+(x)\Big(\frac{f^{2}(u_{n})}{u_{n}^{2}}-1\Big)
v_{n}^{2}
$$
is small as $\varepsilon$ is small.
So $v_n\to 0$ in $E$ which contradicts to  $\|v_n\|=1$. We finish the proof.
\end{proof}

\begin{lemma} \label{lem3.3} 
There exist $k,\rho>0$(small) such that $\inf_{\|u\|=\rho}I_1(u)\geq k\rho^{2}$, 
where $I_1(u):=I(u)+\int_{\mathbb{R}^N}h(x)f(u)$.
 \end{lemma}

\begin{proof}
Due to (G1) and (G2), we have for each $\varepsilon>0$, there exists
 $C_\varepsilon>0$, such that 
$ |g(x,u)|\leq \varepsilon |u|+C_\varepsilon |u|^{p-1}$. 
So it follows from a standard argument by Lemma \eqref{lem2.1}(3),(7) 
and Lemma \ref{lem3.2} that $I_1(u)\geq k\|u\|^2=k\rho^{2}$.
\end{proof}

\begin{lemma} \label{lem3.8} 
For the above $\rho$, $\inf_{\|u\|=\rho}I(u)>0$.
 \end{lemma}
 
\begin{proof}
 By Lemma \ref{lem3.3} and Lemma \ref{lem2.1}-(3), we derive
\begin{align*}
I(u)
&\geq k\|u\|^2-\int_{\mathbb{R}^N}h(x)f(u)\\
&\geq k\|u\|^2-|h|_{L^{2N/(N+2)}}S^{1/2}
\Big(\int_{\mathbb{R}^N}|\nabla u|^2\Big)^{1/2}\\
&\geq k\|u\|^2-|h|_{L^{2N/(N+2)}}S^{1/2}\|u\|\\
&= \|u\|(k\|u\|-|h|_{L^{2N/(N+2)}}S^{1/2})>0\\
\end{align*}
\end{proof}

\begin{lemma} \label{lem3.4} There exists $u_0\neq 0$, such that 
$I(u_0)\leq 0$.
\end{lemma}

\begin{proof}
We have by condition (G3) and Lemma \ref{lem2.1}(3),
$$
\int_{u\neq 0}\frac{G(x,f(tu))}{t^4}=\int_{u\neq 0}
\frac{G(x,f(tu))}{f^4(tu)}\frac{f^4(tu)}{t^4 u^4}u^4\to \infty.
$$
Hence $\lim_{t\to\infty}\frac{I(tu)}{t^4}=-\infty$.
\end{proof}
Since the functional $I$ satisfies the mountain pass geometry, 
the $(C)_c$ sequence exists,
where $c:=\inf_{r\in \Gamma}\max_{t\in [0,1]}I(r(t))$ and 
$\Gamma:=\{r\in C([0,1],E):r(0)=0,r(1)=u_0\}$.

\begin{lemma} \label{lem3.5} 
The $(C)_c$ sequence $(u_n)$ is bounded.
\end{lemma}

\begin{proof}
We employ a similar argument as in \cite[Lemma 3.3]{silva}. 
First we claim
$$
\int_{\mathbb{R}^N}|\nabla u_n|^2+\int_{\mathbb{R}^N}V^+(x)f^2(u_n)\leq C_1.
$$
Indeed, we have 
\begin{gather*}
I(u_n)=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u_n|^2+V(x)f^2(u_n)
-\int_{\mathbb{R}^N}(G(x,f(u_n))+h(x)f(u_n)) \to c\,, \\
\begin{aligned}
 I'(u_n)u_n&=\int_{\mathbb{R}^N}|\nabla u_n|^2 +V(x)f(u_n)f'(u_n)u_n\\
&\quad -\int_{\mathbb{R}^N}(g(x,f(u_n))+h(x))f'(u_n)u_n
\to 0\,.
\end{aligned}
\end{gather*}
Hence 
$$
I(u_n)-\frac{2}{\theta}I'(u_n)u_n=c+o(1).
$$ 
By Lemma \ref{lem2.1}(6),(3) and (G3) we obtain
\begin{align*}
&C_2+C_3\Big(\int_{\mathbb{R}^N}|\nabla u_n|^2\Big)^{1/2}\\
&\geq C_2+(1-\frac{1}{\theta})\int_{\mathbb{R}^N}h(x)f(u_n)\\
&\geq (\frac{1}{2}-\frac{2}{\theta})
 \Big(\int_{\mathbb{R}^N}|\nabla u_n|^2+V^+(x)f^2(u_n)\Big)
 -(\frac{1}{2}-\frac{1}{\theta})\int_{\mathbb{R}^N}V^-(x)f^2(u_n)\\
&\geq (\frac{1}{2}-\frac{2}{\theta})
 \Big(\int_{\mathbb{R}^N}|\nabla u_n|^2+V^+(x)f^2(u_n)\Big)
 -(\frac{1}{2}-\frac{1}{\theta})\int_{\mathbb{R}^N}V^-(x)u_n^2\\
&\geq (\frac{1}{2}-\frac{2}{\theta})
 \Big(\int_{\mathbb{R}^N}|\nabla u_n|^2+V^+(x)f^2(u_n)\Big)
 -(\frac{1}{2}-\frac{1}{\theta})|V^{-}|_{L^{N/2}}S
 \int_{\mathbb{R}^N}|\nabla u_n|^2.
\end{align*}
It follows from (V1) that 
$(\frac{1}{2}-\frac{2}{\theta})
 -(\frac{1}{2}-\frac{1}{\theta})|V^{-}|_{L^{N/2}}S>0$. 
The claim is proved.

To prove that $(u_n)$ is bounded in $E$, we only need to show that 
$\int_{\mathbb{R}^N}V^+(x)u^2_n$ is bounded. 
Due to Lemma \ref{lem2.1}(9), (V1) and the Sobolev embedding theorem, 
there exists $C>0$ such that
$$
\int_{|u_n|\leq 1}V^+(x)u^2_n\leq \frac{1}{C^2}\int_{|u_n|\leq 1}V^+(x)f^2(u_n)
\leq C_3
$$
and
$$
\int_{|u_n|\geq 1}V^+(x)u^2_n\leq C_4\int_{|u_n|\geq 1}u_n^{2^{*}}
\leq C_4\Big(\int_{\mathbb{R}^N}|\nabla u_n|^2\Big)^{2^*/2}\leq C_5.
$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Assume that $(u_n)$ is a $(C)_c$ sequence. Then $(u_n)$ is bounded 
by Lemma \ref{lem3.5}. Going if necessary to a subsequence, 
$u_n\rightharpoonup u$ in $E$. It is obvious that $I'(u)=0$, 
and $u\neq 0$. The proof is complete.
\end{proof}

\section{Proof of Theorem \ref{thm1.2}} \label{s4}

In this section we look for nontrivial critical points of the functional 
$I_1:E\to R$ given by
$$
I_1(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^{2}
+\frac{1}{2}\int_{\mathbb{R}^N}V(x)f^{2}(u)-\int_{\mathbb{R}^N}G(f(u)),
$$
where $G(u):=\int^{u}_{0}g(s)ds$. And we also denote the corresponding 
limiting functional
$$
\tilde{I}_1(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^{2}
+\frac{1}{2}\int_{\mathbb{R}^N}V^+(\infty)f^{2}(u)-\int_{\mathbb{R}^N}G(f(u)).
$$


\begin{lemma} \label{lem4.1} 
If $\{v_n\}\subset E$ is a bounded Palais-Smale sequence for $I_1$ 
at level $c>0$, then, up to a subsequence, $v_n\rightharpoonup v\neq 0$ 
with $I'_1(v)=0$.
\end{lemma}

\begin{proof}
Since $\{v_n\}$ is bounded, going if necessary to a subsequence, 
$v_n\rightharpoonup v$ in $E$. It is obvious that $I'_1(v)=0$. 
If $v\neq 0$, then the proof is complete.

If $v=0$, we claim that $\{v_n\}$ is also a Palais-Smale sequence for 
$\tilde{I}_1$. Indeed,
$$
\tilde{I}_1(v_n)-I_1(v_n)=\int_{\mathbb{R}^N}(V^+(\infty)-V^+(x))f^2(v_n)
+\int_{\mathbb{R}^N}V^-(x)f^2(v_n)\to 0,
$$
by (V1'), Lemma \ref{lem2.1}(3) and $v^2_n\rightharpoonup 0$ in $L^{N/(N-2)}$.
Similarly we derive
\begin{align*}
\sup_{\|u\|\leq 1}|\langle \tilde{I}'_1(v_n)-I'_1(v_n),u\rangle|
&=\sup_{\|u\|\leq 1}\Big|\int_{\mathbb{R}^N}(V^+(\infty)
 -V^+(x))f(v_n)f'(v_n)u\Big| \\
&\quad + \sup_{\|u\|\leq 1}\Big|\int_{\mathbb{R}^N}V^-(x)f(v_n)f'(v_n)u\Big|\to 0.
\end{align*}
In the following we use a similar argument as in \cite[lemma 4.3]{colin1}.
If 
$$
\lim_{n\to\infty}\sup_{y\in\mathbb{R}^N}\int_{B_R(y)}v^2_ndx=0
$$ 
for all $R>0$, then we obtain a contradiction with the fact that 
$I_1(v_n)\to c>0$. So there exist $\alpha>0$, $R<\infty$ and 
$\{y_n\}\subset\mathbb{R}^N$ such that
$$
\lim_{n\to\infty}\int_{B_R(y^n)}v^2_ndx\geq\alpha>0.
$$ 
Denote $\tilde{v}_n(x)=v_n(x+y_n)$, then $\{\tilde{v}_n(x)\}$ is also 
a Palais-Smale sequence for $\tilde{I}_1$. We have that 
$\tilde{v}_n\rightharpoonup \tilde{v}$ and $\tilde{I}_1(\tilde{v})=0$ 
with $\tilde{v}\neq 0$. We obtain
\[
c=\limsup_{n\to\infty}[\tilde{I}(\tilde{v}_n)
 -\frac{1}{2}\tilde{I}'(\tilde{v}_n)\tilde{v}_n]
\geq \tilde{I}(\tilde{v})-\frac{1}{2}\tilde{I}'(\tilde{v})\tilde{v}
=\tilde{I}(\tilde{v}),
\]
by Fatou's lemma.
We could find a path $r(t)\in \Gamma$ such that 
$r(t)(x)>0$ for all $x\in\mathbb{R}^N$, and all $t\in(0,1]$,
$\tilde{\omega}\in r([0,1])$ and 
$\max_{t\in [0,1]}\tilde{I}_1(r(t))=\tilde{I}_1(\tilde{\omega})\leq c$.
Thus $I_1(r(t))<\tilde{I}_1(r(t))$ for all $t\in (0,1]$, and then
$$
c\leq \max_{t\in [0,1]}I_1(r(t))<\max_{t\in [0,1]}\tilde{I}_1(r(t))\leq c,
$$ 
a contradiction.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
The argument is the same as in \cite{colin1}. By Lemmas \ref{lem3.3} 
and \ref{lem3.4}, the functional $I_1$ has a mountain pass geometry. 
So the $(C)_c$-sequence $\{u_n\}$ exists, where 
$c:=\inf_{r\in \Gamma}\max_{t\in [0,1]}I_1(r(t))$ and 
$\Gamma:=\{r\in C([0,1],E):r(0)=0,I_1(r(1))<0\}$. 
It follows from Lemma \ref{lem3.5} that $\{u_n\}$ is bounded. 
Hence $\{u_n\}$ is a bounded Palais-Smale sequence for $I_1$ at level $c>0$.
Due to Lemma \ref{lem4.1}, we have $I'_1(v)= 0$ and $v\neq 0$.
\end{proof}

\subsection*{Acknowledgements}
The first author would like to thank Andrzej Szulkin for valuable
suggestions about the draft of the paper. The authors are supported
by NSFC 11171047.

\begin{thebibliography}{99}

\bibitem{alves} M. J. Alves, P.C. Carri\~{a}o, O. H. Miyagaki;
\emph{Non-autonomous perturbations for a class of quasilinear elliptic 
equations on $\mathbb{R}$}, J. Math. Anal. Appl. 344 (2008), 186--203.

\bibitem{colin} M. Colin;
\emph{Stability of standing waves for a quasilinear Schr\"odinger equation 
in space dimension 2}, Adv. Diff. Eq. 8 (2003), 1--28.

\bibitem{colin1} M. Colin, L. Jeanjean;
\emph{Solutions for a quasilinear Schr\"{o}dinger equation: a dual approach},
 Nonl. Anal. 56 (2004), 213--226.

\bibitem{deng} Y. B. Deng, S. J. Peng, J. X. Wang;
\emph{Infinitely many sign-changing solutions for quasilinear Schr\"{o}dinger 
equations in $\mathbb{R}^N$ }, Comm. Math. Sci. 9 (2011), 859--878.

\bibitem{ding} Y. H. Ding, A. Szulkin;
\emph{Bound states for semilinear Schr\"{o}dinger equations with sign-changing
 potential}, Calc. Var. 29 (2007), 397--419.

\bibitem{severo} J. M. do  \'{O}, U. Severo;
\emph{Quasilinear Schr\"{o}dinger equations involving concave and convex 
nonlinearities}, Comm. Pure Appl. Anal. 8 (2009), 621--644.

\bibitem{severo1} J. M. do \'{O}, U. Severo;
\emph{Solitary waves for a class of quasilinear Schr\"o\-dinger equations 
in dimension two}, Calc. Var. 38 (2010), 275--315.

\bibitem{fang} X. D. Fang, A. Szulkin;
\emph{Multiple solutions for a quasilinear Schr\"{o}dinger equation}, 
J. Diff. Eq. 254 (2013), 2015--2032.

\bibitem{wang1} J. Q. Liu, Y. Q. Wang, Z. Q. Wang;
\emph{Soliton solutions for quasilinear Schr\"{o}dinger equations, II}, 
J. Diff. Eq. 187 (2003), 473--493.

\bibitem{wang2} J. Q. Liu, Y. Q. Wang, Z. Q. Wang;
\emph{Solutions for quasilinear Schr\"{o}dinger equations via the Nehari method}, 
Comm. PDE 29 (2004), 879--901.

\bibitem{wang3} J. Q. Liu, Z. Q. Wang;
\emph{Soliton solutions for quasilinear Schr\"{o}dinger equations, I}, 
Proc. Amer. Math. Soc. 131 (2003), 441--448.

\bibitem{poppenberg} M. Poppenberg, K. Schmitt, Z. Q. Wang;
\emph{On the existence of soliton solutions to quasilinear Schr\"odinger 
equations}, Calc. Var. 14 (2002), 329--344.

\bibitem{rabinowitz} P. H. Rabinowitz;
\emph{Minimax Methods in Critical Point Theory with Applications to 
Differential Equations}, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. 
Math. Soc., Providence, RI, 1986.

\bibitem{silva} E. A. Silva, G. G. Vieira;
\emph{Quasilinear asymptotically periodic Schr\"{o}din\-ger equations 
with subcritical growth}, Nonl. Anal. 72 (2010), 2935--2949.

\bibitem{szulkin} A. Szulkin, M. Willem;
\emph{Eigenvalue problems with indefinite weight}, 
Studia Math. 135 (1999), no.2, 191--201.

\bibitem{zou} Y. J. Wang, W. M. Zou;
\emph{Bound states to critical quasilinear Schr\"{o}dinger equations}, 
Nonl. Diff. Eq. Appl. 19 (2012), 19--47.

\bibitem{willem} M. Willem;
\emph{Minimax Theorems}, Birkh\"{a}user, Boston, 1996.

\bibitem{zhang} Z. H. Zhang, R. Yuan;
\emph{Homoclinic solutions for some second order non-autonomous Hamiltonian 
systems without the globally superquadratic condition}, 
Nonl. Anal. 72 (2010), 1809--1819.

\end{thebibliography}

\end{document}