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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 91, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/91\hfil Quasilinear Schr\"odinger equations]
{Existence of solutions to  quasilinear Schr\"odinger equations
with indefinite potential}

\author[Z. Shen, Z. Han \hfil EJDE-2015/91\hfilneg]
{Zupei Shen, Zhiqing Han}

\address{Zupei Shen \newline
School of Mathematical Sciences, Dalian University of Technology,
Dalian 116024,  China}
\email{pershen@mail.dlut.edu.cn}

\address{Zhiqing Han (corresponding author)\newline
School of Mathematical Sciences, Dalian University of Technology,
Dalian 116024,  China}
\email{hanzhiq@dlut.edu.cn Phone(Fax) +86 41184707268}

\thanks{Submitted October 24, 2014. Published April 10, 2015.}
\subjclass[2000]{37J45, 58E05, 34C37, 70H05}
\keywords{Quasilinear Schr\"odinger equation; local linking;
fountain theorem; \hfill\break\indent indefinite potential}

\begin{abstract}
 In this article, we study the existence and multiplicity of solutions
 of the  quasilinear Schr\"odinger equation
 $$
 -u''+V(x)u-( | u| ^2)''u=f(u)
 $$
 on $\mathbb{R}$, where the potential $V$ allows sign changing and the
 nonlinearity satisfies conditions weaker than the classical
 Ambrosetti-Rabinowitz condition. By a  local linking theorem and the fountain
 theorem, we  obtain the existence and multiplicity of solutions for the equation.
\end{abstract}

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

\section{Introduction}

We study the existence and multiplicity of solutions for the quasilinear elliptic
equation
\begin{equation}
-u''+V(x)u-( | u| ^2)''u=f(u), \quad  x\in\mathbb{R}. \label{QS}
\end{equation}
Solutions of the equation are related to  standing wave solutions
for quasilinear Schr\"odinger equation of the form
\begin{equation}
i\partial _{t}z=-z''+\tilde{V}(x)z-(| z|
^2)''z-\tilde{f}(|
z| ^2)z  \label{s}
\end{equation}
which arises in various fields of physics, like the theory of
superfluids or in dissipative quantum mechanics, plasma physics,
fluid mechanics and  in the
theory of Heisenberg ferromagnets, etc.  For further physical motivations and a
more complete list of references, we refer to
\cite{JiaQuan2002,Liu2012,Poppenberg2002} and the references therein.

As far as we know, the first
existence result for equation \eqref{QS}  by variational methods  is due
to \cite{Poppenberg2002}, where by a constrained minimization argument
the authors proved the existence of a positive ground state solution with
an unknown Lagrange multiplier $\lambda$ in front
of the nonlinear term. Ambrosetti and Wang ~\cite{Wang2003} considered the
existence of positive solutions of perturbation to the equation with
a particular nonlinearity $g(u)=u^p$. Alves et al \cite{MR2838282},
considered the existence and concentration of positive solutions as
 $\epsilon\to 0$ for a related equation with $\epsilon^2$.
Some related problems on $\mathbb{R}$ are also considered in \cite{MR2416300}.

There is also much work devoting to the corresponding high dimensional equation;
e.g. see  \cite{JiaQuan2002,LiuJia-Quan2004,Liu2003,Liu2012}.
The solutions of  equation \eqref{QS} correspond to the  critical
points of the  functional on $H^1(\mathbb{R})$:
\begin{equation}
\Phi (u)=\frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
+\int_{\mathbb{R}}{u'}^2u^2dx-\int_{\mathbb{R}}F(u)dx , \label{1}
\end{equation}
where $F(t)=\int^t_0f(s)ds$.
All the above mentioned  papers require that the
potential $V(x)$ is positive. So that the energy functional $\Phi $
possesses the mountain pass geometry. Therefore, the mountain pass lemma can be
applied. In \cite{zhang1}, the authors consider the case which the
potential $V(x)$ allows sign-change. However, in order to satisfy the conditions
of mountain pass theory, they need additional conditions on nonlinearity.
The similar assumptions are added in~\cite{zhang2} (See Remark ).
The aim of the paper is to investigate equation \eqref{QS} where the potential
$V(x)$ can be  sign changing  and the nonlinearity does not need to
satisfy Ambrosetti-Rabinowitz condition. The term
$\int_{\mathbb{R}}{u'}^2u^2dx$ in \eqref{1} is homogeneous of order 4 and
non-convex, it prevents the  linking geometric structure of the energy
functional under our assumption. Inspired by the recent work of
Chen and Liu \cite{Chen} we make use of the local linking theory to overcome this
difficulty.
To state our main results, we list the assumptions on $f$ and $V$ as follows.
\begin{itemize}
\item[(V1)]  The potential $V(x)\in C(\mathbb{R})$ is bounded from below and
$\mu (V^{-1}(-\infty ,M))<\infty $ for every $M>0$.
\item[(F1)] There exists $C>0$ and $p>2$ such that
\begin{equation*}
| f(t)| \leq C| t| ^{p-1}  \quad\text{for all }  t \in \mathbb{R}. \label{3}
\end{equation*}

\item[(F2)] $4F(t)\leq f(t)t$   and
\begin{equation*}
\lim_{t\to \infty }\frac{F(t)}{t^{4}}=+\infty \quad\text{for all }
t \in \mathbb{R}. \label{4}
\end{equation*}
\end{itemize}
We are now ready to state our results.


\begin{theorem} \label{thm1.1}
 Suppose that {\rm (V1), (F1)--(F2)} are satisfied and $f$ is
odd. Then  equation \eqref{QS} has a sequence of solutions such that
$\Phi (u_{k})\to +\infty$.
\end{theorem}

\begin{theorem} \label{thm1.2}
 Suppose that  {\rm (V1), (F1)--(F2)} are satisfied. Then
equation \eqref{QS} has at least one nontrivial solution.
\end{theorem}

\begin{remark} \label{rmk1} \rm
In \cite{zhang1,zhang2}, the authors need assumptions (F2) and
\begin{itemize}
\item[(G2)]
$\widetilde{F}(x, u) :=\frac{1}{4}f(x, u)u- F(x, u)\geq 0$,
and there exist $c_0>0$ and $\sigma > max\{1,\frac{2N}{N+2}\}$
such that
$$
|F(x,u)|^\sigma \leq c_0|u|^{2\sigma}\widetilde{F}(x, u)
$$
for all $(x, u)\in \mathbb{R}^N$.
\end{itemize}
In this paper, $(G_2)$ is not needed.
\end{remark}

Throughout this paper, the letters $C$ and
$C_i$ denote positive constants, which may be different from place to place.
The usual norm in $L^p(\mathbb{R})$ with $1\leq p\leq+\infty$ is
denoted by $|\cdot|_p$

\section{Preliminaries}

To overcome the non-compactness of the embedding $H^{1}(\mathbb{R}
)\hookrightarrow L^2(\mathbb{R})$, we consider a linear subspace $X$ of
$H^{1}(\mathbb{R})$:
\begin{equation*}
X:=\{u\in H^{1}(\mathbb{R}):\int_{\mathbb{R}}\bar{V}(x)u^2dx<\infty\}
\end{equation*}
equipped with the inner product
\begin{equation*}
\langle u,v\rangle =\int_{\mathbb{R}}{u'}{v'}+\bar{V}uvdx
\end{equation*}
and the corresponding norm $\| u\| =\langle u,v\rangle ^{1/2}$, where
$\bar{V}=V(x)+m>1$ for a fixed positive number $m$.
It is well known that the  imbedding $X\hookrightarrow L^2(\mathbb{R})$
is compact under the condition (V1).
See\cite{B-W}. Therefore, the eigenvalues of the operator
\begin{equation*}
S:=-\Delta +V
\end{equation*}
can be numbered as
\begin{equation*}
-\infty <\lambda _{1}\leq \lambda _{2}\leq \lambda _{3}\dots ,\lambda
_{l}\to \infty
\end{equation*}
and the  corresponding eigenfunctions are denoted by
$\phi _{1},\text{ }\phi _{2}\dots$.
We assume that $0\in (\lambda _{l},\lambda _{l+1})$ for some $l>1$. Let
\begin{equation*}
X^{-}=\operatorname{span}\{\phi _{1}\text{,}\dots , \phi _{l}\},\quad
X^{+}=(X^{-})^{\perp }.
\end{equation*}
Then $X^{-}$ and $X^{+}$ are the negative and  positive spaces of
the quadratic form
\begin{equation*}
Q(u)=\frac{1}{2}\int_{\mathbb{R}}{u'(x)}^2+V(x)u^2(x)dx.
\end{equation*}.
It is well known that there is a positive constant $\alpha>0$ such
that
\begin{equation}
\pm Q(u)\geq\alpha\| u\| ^2,\quad u\in X^{\pm }.
\label{1.00}
\end{equation}
Let
\[
J(u)=\frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
-\int_{\mathbb{R}}F(u)dx,\quad
I(u)=\int_{\mathbb{R}}{u'}^2u^2dx.
\]
 Then
\begin{equation*}
\Phi (u)=J(u)+I(u).
\end{equation*}
By the continuous imbedding $H^{1}(\mathbb{R})\hookrightarrow L^{\infty }(
\mathbb{R})$ and $X\hookrightarrow H^{1}(\mathbb{R})$,  $I$ is well
defined on $X$ and
\begin{equation}
\ | I(u)| \leq | u| _{\infty}^2\| u\| _{H^{1}}^2\leq C\| u\| ^{4} \quad
\text{for  all } u\in X. \label{2.7}
\end{equation}
To verify that the functional $\Phi$ is  $C^{1}$, it is sufficient to
 prove  this for  $I(u)$.


\begin{lemma} \label{lem2.1}
 $I(u)$ belongs to $C^{1}$ in $X$.
\end{lemma}

The proof of the above lemma is similar to that of  \cite[Lemma 1]{Poppenberg2002}.
We omit it  here.
From the above discussions, the functional $\Phi (u)$ is a $C^{1}$ functional
 with derivative  given by
\begin{equation}
\begin{aligned}
(\Phi '(u),v) &= \int_{\mathbb{R}}{u'}{v'}+\bar{V} uv dx
+\int_{\mathbb{R}}2{u'}^2uv+2u^2{u'}{v'}dx-\int_{\mathbb{R}
}g(u)v dx   \\
&= \langle u,v\rangle +\int_{\mathbb{R}}2{u'}^2uv+2u^2{u'}{v'}dx
-\int_{\mathbb{R}}g(u)v dx,
\end{aligned} \label{5}
\end{equation}
where $g(t)=f(t)+mt$. To prove our main results, we need to introduce
some definitions and theorems.
\smallskip

\noindent \textbf{Definition.}
We say that $\Phi \in C^{1}(X)$ satisfies condition (PS) if any sequence
 ($u_n$)$ \subset X$ such that
$$
\Phi(u_n)\to c, \quad \Phi'(u_n)\to 0
$$
has a convergent subsequence.

 For the proof of Theorem \ref{thm1.1}, we use the following fountain theorem by
Bartsch \cite{Bartsch1993}.
 For $k=1,2\dots $, let
\begin{equation}
Y_{k}=\operatorname{span}\{\phi _{1},\dots \phi _{k}\},\quad
Z_{k}=\overline{\operatorname{span}\{\phi _{k},\phi _{1+k}\dots \}}.  \label{2.11}
\end{equation}

\begin{theorem} \label{thm2.3}
Assume that the even functional $\Phi \in C^{1}(X)$  satisfies the (PS) condition,
if there exists $k_{0}>0$ such that for $k\geq k_{0}$ there exists
$\rho _{k}>r_{k}>0$ such that
\begin{itemize}
\item[(i)] $b_{k}=\inf_{u\in Z_{k},\|u\| =r_{k}}\Phi (u)\to +\infty $
 as $k\to \infty$,

\item[(ii)] $\alpha _{k}=\max_{u\in Y_{k},\|u\| =\rho _{k}}\Phi (u)\leq 0$.
\end{itemize}
Then $\Phi $ has a sequence of critical points $\{u_{k}\}$ such that
$\Phi(u_{k})\to +\infty$.
\end{theorem}

For the proof of Theorem \ref{thm1.2}, we will use the local linking theorem.
Recall that the definition of local linking at 0 with respect to the
direct sum decomposition $X=X^{+}\oplus X^{-}$, if there is $\rho >0$ such
that for $u\in X^{-}$
\begin{equation}
\begin{gathered}
\Phi (u)\leq 0,\quad \text{for }u\in X^{-},\; \| u\| \leq \rho \\
\Phi (u)\geq 0,\quad\text{for }u\in X^{+}, \; \| u\|\leq \rho.
\end{gathered}  \label{2.8}
\end{equation}
Next, we consider two sequences of finite dimensional subspaces
\begin{equation*}
X_{0}^{\pm }\subset X_{1}^{\pm }\subset \dots \subset X^{\pm }
\end{equation*}
such that
\begin{equation*}
X^{\pm }=\overline{\cup _{n\in \mathbb{N}}X_{n}^{\pm }}.
\end{equation*}
For multi-index $\alpha =(\alpha ^{-},\alpha ^{+})\in \mathbb{N}^2$, we
set $\ X_{\alpha }=X_{\alpha }^{-}\oplus X_{\alpha }^{+}$
 and denote by
$\Phi _{\alpha }$ the restriction of $\Phi $ on X$_{\alpha }$.
A sequence $\{\alpha _{n}\}\subset \mathbb{N}^2$ is admissible if,
for any $\alpha \in \mathbb{N}^2$, there is $m\in \mathbb{N}$ such that
$\alpha \leq \alpha_{n}$ for $n\geq m$, where for
$\alpha ,\beta \in \mathbb{N}^2$,
$\alpha \leq \beta $ means $\alpha ^{\pm }\leq \beta ^{\pm }$.
Obviously, if $\{\alpha _{n}\}$ is admissible, then any subsequence of
$\{\alpha _{n}\}$ is also admissible.
\smallskip

\noindent\textbf{Definition.}
We say that $\Phi \in C^{1}(X)$ satisfies condition (C)$^{\ast}$ if, whenever
$\{\alpha _{n}\}\subset \mathbb{N}^2$ admissible, any
sequence $\{u_{n}\}\subset X$ such that
\begin{equation}
u_{n}\in X_{\alpha _{n}}, \quad\sup_{n}\Phi (u_{n})<\infty , \quad
(1+\| u_{n}\|)\| \Phi _{_{\alpha _{n}}}'(u_{n})\| \to 0  \label{2.9}
\end{equation}
contains a subsequence which converges to a critical point of $\Phi $.

\begin{theorem}[Local linking theorem \cite{Luan 2005}] \label{thm2.4}
Suppose that $\Phi \in C^{1}(X)$ has a local linking at
$0$, $\Phi $ satisfies (C)$^{\ast }$, $\Phi $ maps bounded sets into
bounded sets and for every $m\in \mathbb{N}$,
\begin{equation}
\Phi (u)\to -\infty \quad \text{as }\| u\| \to
\infty ,\;  u\in X^{-}\oplus X_{m}^{+}.  \label{2.10}
\end{equation}
Then $\Phi $ has a nontrivial critical point.
\end{theorem}

\section{Proofs of Theorems \ref{thm1.1} and \ref{thm1.2}}


It is reasonable to write the functional $\Phi $ in a form
in which the quadratic part is $\| u\| ^2$. Let $g(t)=f(t)+mt$.
By (F1)--(F2), it is known that $G(t)$ satisfies the following properties
\begin{gather}
G(t)\leq \frac{t}{4}g(t)+\frac{m}{4}t^2,\label{10} \\
\lim_{| t|\to \infty }\frac{G(t)}{t^{4}}=+\infty, \label{11}
\end{gather}
and hence there is a $\Lambda >0$ such that
\begin{equation}
G(t)\geq -\Lambda t^{4} \quad \text{for all }t\in \mathbb{R}.\label{14}
\end{equation}

\begin{lemma} \label{lem3.1}
Suppose that {\rm (V1), (F1)--(F2)} are satisfied, then $\Phi $
satisfies the (PS) condition.
\end{lemma}

\begin{proof}
Suppose that $\{u_{n}\}$ is a (PS) sequence. We claim that $\{u_{n}\}$ is bounded.
By contradiction, we may assume that  $\|u_{n}\| \to \infty $. Using \eqref{5}
and \eqref{10},we have
\begin{equation}
\begin{aligned}
4\sup_{n}\Phi (u_{n})+\| u_{n}\|
&\geq  4\Phi (u_{n})-(\Phi '(u_{n}),u_{n})   \\
&= \| u_{n}\| ^2+\int_{\mathbb{R}}g(u_{n})u_{n}-4G(u_{n})dx   \\
&\geq \| u_{n}\| ^2-m\int_{\mathbb{R}}u_{n}^2dx.
\end{aligned}\label{3.1}
\end{equation}
By \eqref{3.1}, we  obtain
\begin{equation}
\| u_n\|=O(|u_{n}|_2)\label{*}.
\end{equation}
Let $v_{n}=\| u_{n}\| ^{-1}u_{n}$. Up to a subsequence, by the
compact embedding $X\hookrightarrow L^2(\mathbb{R})$ we can assume that
\begin{equation*}
v_{n}\rightharpoonup v\text{ in }X, \quad
v_{n}\to v\text{ in } L^2(\mathbb{R}),\quad
v_{n}(x)\to v(x) \text{ a.e in } \mathbb{R}.
\end{equation*}
By \eqref{*}, we have
\begin{equation*}
|v_{n}|_2\geq \frac{|u_{n}|_2}{c|u_{n}|_2}=\frac{1}{c}>0
\end{equation*}
for some positive constant $c>0$.
Therefor  $v\neq 0$.
Using \eqref{14}, we obtain
\begin{align*}
\int_{\mathbb{R}}\frac{G(u_{n})}{\| u_{n}\| ^{4}}dx
&=  \int_{v=0}\frac{G(u_{n})}{\| u_{n}\| ^{4}}dx
 +\int_{v\neq0}\frac{G(u_{n})}{\| u_{n}\| ^{4}}dx\\
&= \int_{v=0}\frac{G(u_{n})}{u_{n}^{4}}v_{n}^{4}dx
 +\int_{v\neq 0}\frac{G(u_{n})}{u_{n}^{4}}v_{n}^{4}dx\\
&\geq -\Lambda \int_{v=0}v_{n}^{4}\text{ }dx
 + \int_{v\neq 0}\frac{G(u_{n})}{u_{n}^{4}}v_{n}^{4}dx\\
&= I_1+I_2.
\end{align*}
Obviously,
$I_1\geq -c >-\infty$.
For $x\in \{x\in\mathbb{R} |v\neq 0\}$, we have
$| u_{n}| \to \infty $. By \eqref{11}, we obtain
\begin{equation*}
\frac{G(u_{n})}{\| u_{n}\| ^{4}}=\frac{G(u_{n})}{u_{n}^{4}}v_{n}^{4}\to +\infty.
\end{equation*}
By Fatou's lemma,
$I_2\to+\infty$.
Then we have
\begin{equation}
\int_{\mathbb{R}}\frac{G(u_{n})}{\| u_{n}\| ^{4}}dx\to +\infty.\label{***}
\end{equation}
On the other hand
\begin{align*}
\int_{\mathbb{R}}{G(u_{n})}dx
&= \frac{1}{2} \| u_{n}\| ^2+\int_{\mathbb{R}}{u_{n}'}^2u_{n}^2dx-\Phi (u_{n})\\
&\leq \frac{1}{2} \| u_{n}\| ^2+c\| u_{n}\| ^{4}+C.
\end{align*}
Then we have
\begin{equation}
\int_{\mathbb{R}}{G(u_{n})}dx =O(\| u_{n}\| ^{4}),
\end{equation}
a contradiction to \eqref{***}. So $\{u_{n}\}$ is bounded.


Next, we  show that such sequence $\{u_{n}\}$ has a subsequence
converging to a critical point of $\Phi $. Because $\{u_{n}\}$ is bounded in
 $X$, we may assume $u_{n}\rightharpoonup u$ in $X$. Since the imbedding
$X\hookrightarrow L^{p}(\mathbb{R})$ is compact, we have $u_{n}\to u$
in $L^{p}(\mathbb{R})$.
By a simple computation, we have
\begin{equation*}
\Phi '(u_{n})(u_{n}-u)=\| u_{n}-u\| ^2
+2\int_{\mathbb{R}}{u'}^2(u_{n}-u)^2
+2\int_{\mathbb{R}}u^2({u'}_{n}-{u'})^2
-\int_{\mathbb{R}}g(u_{n})(u_{n}-u).
\end{equation*}
By condition (F1) and   Holder's inequality, we have
\begin{equation*}
\int_{\mathbb{R}}f(u_{n})(u_{n}-u)
\leq C| u|_p^{p-1} | u_{n}-u| _p.
\end{equation*}
Since $u_{n}\to u$ in $L^{p}(\mathbb{R})$ and $p\geq 1$, we have
\begin{equation}
\int_{\mathbb{R}}f(u_{n})(u_{n}-u)dx\to 0\quad \text{as }n\to \infty.  \label{e1}
\end{equation}
Similarly,
\begin{equation}
\int_{\mathbb{R}}mu_{n}(u_{n}-u)dx\to 0\quad \text{as }n\to \infty.  \label{e2}
\end{equation}
By \eqref{e1} and \eqref{e2}, we obtain
\begin{equation}
\int_{\mathbb{R}}g(u_{n})(u_{n}-u)\to 0\quad \text{as }n\to \infty.  \label{e3}
\end{equation}
 By the assumptions we have
\begin{equation}
\Phi '(u_{n})(u_{n}-u)=o(\| u_{n}-u\| ).\label{e}
\end{equation}
From \eqref{e3} we obtain
\begin{equation*}
\Phi '(u_{n})(u_{n}-u)=\| u_{n}-u\| ^2
+2\int_{\mathbb{R}}{u'}^2(u_{n}-u)^2+2\int_{\mathbb{R}}u^2({u'}_{n}-{u'})^2+o(1).
\end{equation*}
Hence we  obtain $\| u_{n}-u\| \to 0$ as $n\to \infty$. This completes the proof.
\end{proof}

\begin{lemma} \label{lem3.2}
Under assumptions {\rm (V1), (F1)--(F2)}, the
functional $\Phi $ has a local linking at 0 with respect to the decomposition $
X=X^{+}\oplus X^{-}$.
\end{lemma}

\begin{proof}
By (F1), there exists a $ C>0$
such that
\begin{equation}
| F(u)| \leq C| u| ^{p}.  \label{3.2}
\end{equation}
Using \eqref{1.00}, \eqref{2.7}, for $u\in X^{-}$, there exists a
$\delta >0$. Then we have
\begin{equation}
\begin{aligned}
\Phi (u)
&= \frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
 +\int_{\mathbb{R}}{u'}^2u^2dx-\int_{\mathbb{R}}F(u)dx   \\
&\leq \frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
 +C\| u\| ^{4}+C| u| _p^{p}  \\
&\leq -\alpha\| u\| ^2+C\| u\| ^{4}+C_{1}| u| _p^{p}\\
&\leq -\delta \| u\| ^2+C_{}\| u\|^{4}+C_{1}\| u\| ^{p}.
\end{aligned}  \label{3.4}
\end{equation}
If $u\in X^{+}$, then there exists  $\xi>0$ such that
\begin{equation}
\begin{aligned}
\Phi (u)
&= \frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
+\int_{\mathbb{R}}{u'}^2u^2dx-\int_{\mathbb{R}}F(u)dx   \\
&\geq \xi \| u\| ^2-C_{1}\| u\|^{p}.
\end{aligned}  \label{3.5}
\end{equation}
By \eqref{3.4} and \eqref{3.5}, there exists  $0<\rho <1$, such that
\begin{gather*}
\Phi (u)\leq 0\quad \text{for }u\in X^{-}, \| u\| \leq \rho, \\
\Phi (u)\geq 0\quad \text{for }u\in X^{+}, \| u\| \leq \rho.
\end{gather*}
This completes the proof.
\end{proof}


\begin{lemma} \label{lem3.3}
 Let $Y$ be a finite dimensional subspace of $X$. Then $\Phi$ is anti-coercive
on $Y$; that is,
\begin{equation*}
\Phi (u)\to -\infty \quad \text{as }\|u\| \to \infty \;u\in Y.
\end{equation*}
\end{lemma}

\begin{proof}
 A similar lemma was proved in \cite{Chen}, we sketch the proof here for the
reader's convenience.
 If the conclusion were not true, we can choose $\{u_{n}\}\subset Y$ and
$\varsigma \in\mathbb{R}$ such that
\begin{equation}
\| u_{n}\| \to \infty,\quad  \Phi(u_{n})\geq \varsigma.  \label{3.6}
\end{equation}
Let $v_{n}=\| u_{n}\| ^{-1}u_{n}$. Since $\dim Y<\infty $,
up to a subsequence, we have
\begin{equation*}
\| v_{n}-v\| \to 0,\quad v_{n}(x)\to v(x)\quad \text{a.e. }\mathbb{R}
\end{equation*}
for some $v\in Y$ with $\| v\| =1$. If $v(x)\neq 0$, we have
$| u_{n}(x)| \to \infty $.
Using \eqref{2.7} and \eqref{***}, we deduce
\begin{equation*}
\Phi (u_{n})\leq \| u_{n}\| ^{4}\Big(\frac{1}{2\|
u_{n}\| ^2}+C-\int_{\mathbb{R}}\frac{G(u_{n})}{\| u_{n}\| ^{4}}dx\Big)\to -\infty,
\end{equation*}
a contradiction with \eqref{3.6}.
\end{proof}


\begin{lemma} \label{lem3.4}
 Suppose {\rm, (V1), (F1)-(F2)} are satisfied. Then $\Phi $
satisfies condition (C)$^{\ast }$.
\end{lemma}

This proof is similar to the Lemma \ref{lem3.1} and is omitted here.
See also \cite{Chen}.


\begin{proof}[Proof of Theorem \ref{thm1.1}]
 It suffices to verify that
\begin{itemize}
\item[(i)] $b_{k}=\inf_{u\in Z_{k},\| u\|=r_{k}}\Phi (u)\to +\infty $
 as $k\to \infty$,

\item[(ii)] $\alpha _{k}=\max_{u\in Y_{k},\|u\| =\rho _{k}}\Phi (u)\leq 0$.
\end{itemize}

(i) We claim that for any $2\leq p$, we have
\begin{equation}
\beta _{k}:=\sup_{u\in Z_{k},\| u\| =1}\|
u\| _{L^{p}}\to 0,\quad \text{as }k\to \infty. \label{3.01}
\end{equation}
If the conclusion were not true, we may assume that
$\beta _{k}\to \beta >0$ as $k\to \infty $. Then there exists a
$u_{k}\in Z_{k}$ with $\| u_{k}\| =1$ and $\| u_{k}\|_p\geq \frac{\beta }{2}$
for large $k$. By the Parseval equality we have
\begin{align*}
\langle u,u_{k}\rangle
&= \big| \langle \sum_{j=k}^{\infty }\alpha
_{j}\phi _{j},u_{k}\rangle \big| \\
&\leq \| \sum_{j=k}^{\infty }\alpha _{j}\phi _{j}\|\,\| u_{k}\| \\
&= \Big(\sum_{j=k}^{\infty }\alpha _{j}^2\Big)^{1/2}\to 0
\end{align*}
where $\langle ,\rangle $ denotes the inner product in $X$. Using the
Riesz-Frechet representation theorem, we obtain that $u_{k}\rightharpoonup 0$
and thus $u_{k}\to 0$ in $L^{p}$. This is a contradiction.
For $u\in Z_{k}$ with $\| u\| =r_{k}$, for enough small $\epsilon $,
\begin{align*}
\Phi (u)
&= \frac{1}{2}\int_{\mathbb{R}}{u'}^2+V(x)u^2dx
+\int_{\mathbb{R}}{u'}^2u^2dx-\int_{ \mathbb{R}}F(u)dx \\
&\geq  k\| u\| ^2-\int_{\mathbb{R}}F(u)dx \\
&\geq  k\| u\| ^2-C| u|_p^{p} \\
&\geq  k\| u\| ^2-C\beta _{k}^{p}\|u\| ^{p}.
\end{align*}
Choosing $r_{k}=\beta _{k}^{-1}$,  we have
\begin{equation*}
\Phi (u)\geq k\beta _{k}^{-2}-C\to +\infty.
\end{equation*}
This proves (i). \smallskip

(ii) Since $\dim Y_{k}<\infty $, using  Lemma \ref{lem3.3}, we have
\begin{equation*}
\Phi (u)\to -\infty \quad \text{for }u\in Y_{k}\text{ and }\rho_{k}\to \infty.
\end{equation*}
Then we obtain
\begin{equation*}
\alpha _{k}=\max_{u\in Y_{K},\| u\| =\rho _{k}}\Phi(u)\leq 0.
\end{equation*}
This completes  the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 In Lemmas \ref{lem3.2} and \ref{lem3.4}, we see that $\Phi $
satisfies  condition (C)$^{\ast }$, and has a local linking at $0$.
Since $\dim (X^{-}\oplus X_{m}^{+})<\infty$,
By Lemma \ref{lem3.3}, we have $\Phi (u)\to -\infty $, as
$\| u\| \to \infty $, $u\in X^{-}\oplus X_{m}^{+}$. By Theorem \ref{thm2.4},
equation \eqref{QS} has at least one nontrivial solution.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by NSFC (11171047), NSFC (11171204) and
GDNSF (S2012010010038).

\begin{thebibliography}{99}

\bibitem{MR2838282} C.~O. Alves, O.~H. Miyagaki,  S.~H.~M. Soares;
\emph{On the existence   and concentration of positive solutions to a class
of quasilinear elliptic   problems on  $\mathbb{R}$}, Math. Nachr., 284 (2011),
  pp. 1784--1795.

\bibitem{MR2416300} 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), pp.~186--203.

\bibitem{Bartsch1993} T.~Bartsch;
\emph{Infinitely many solutions of a symmetric {D}irichlet
  problem}, Nonlinear Anal., 20 (1993), pp.~1205--1216.

\bibitem{B-W} T. Bartsch, Z. Q. Wang;
\emph{Existence and multiplicity results for some superlinear elliptic
problems on $\mathbb{R}^N$}, Comm.
Partial Differential Equations 20 (1995), pp.~ 1725-1741.

\bibitem{Chen} H.~Chen, S.~Liu;
\emph{Standing waves with large frequency for
  4-superlinear Schr\"{o}dinger-Poisson systems},
Ann. Mat. Pura Appl. (4) 194 (2015), no. 1, pp. 43-53.

\bibitem{JiaQuan2002}  J. Q. Liu, Z. Q. Wang;
\emph{Soliton solutions for quasilinear   Schr\"{o}dinger equations. I},
 Proc. Amer. Math. Soc., 131 (2002),   pp. 441--448.

\bibitem{Luan2005} S. Luan, A. Mao;
\emph{Periodic solutions for a class of non-autonomous Hamiltonian systems.}
Nonlinear Anal. 61 (2005), no. 8, pp. 1413-1426.

\bibitem{Liu2003} J. Q. Liu, Y. Q. Wang, Z. Q. Wang;
\emph{Soliton solutions for   quasilinear Schr\"{o}dinger equations, II},
J. Differential Equations, 187  (2003), pp. 473--493.

\bibitem{Liu2012} J. Q. Liu, Z. Q. Wang, Y. X. Guo;
\emph{Multibump solutions for  quasilinear elliptic equations},
Journal of Functional Analysis, 262 (2012),   pp. 4040--4102.

\bibitem{LiuJia-Quan2004} J. Q. Liu, Y. Q. Wang, Z. Q. Wang;
\emph{Solutions for quasilinear  Schr\"{o}dinger  equations via Nehari method},
Comm. Partial Differential Equations 29 (2004) pp. 879--901.

\bibitem{Poppenberg2002} M.~Poppenberg, K.~Schmitt, Z. Q. Wang;
\emph{On the existence of soliton solutions to quasilinear
Schr\"{o}dinger equations}, Calc. Var. Partial Differential Equations, 344 (2002),
  pp. 329--344.

\bibitem{Wang2003} A.~Ambrosrttiz, Z. Q.~Wang;
\emph{Positive solutions to a class of quasilinear elliptic equations
on $\mathbb{R}$}, Discrete Contin. Dynam. Systems, 9 (2003), pp. 55--68.

\bibitem{zhang1} J.~Zhang, X.~Tang, W.~Zhang;
\emph{Infinitely many solutions of quasilinear Schr\"{o}dinger equation with
sign-changing potential}, J. Math. Anal. Appl. 420 (2014), no. 2, pp. 1762-1775.

\bibitem{zhang2} J.~Zhang, X.~Tang, W.~Zhang;
\emph{Existence of infinitely many solutions for a quasilinear elliptic equation}
 Appl. Math. Lett. 37 (2014), pp. 131-135.

\end{thebibliography}

\end{document}