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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 147, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/147\hfil Modified nonlinear  Schr\"odinger equation]
{Existence of infinitely solutions for a modified nonlinear
 Schr\"odinger equation via dual approach}

\author[X. Zhang,  L. Liu,  Y. Wu, Y. Cui \hfil EJDE-2018/147\hfilneg]
{Xinguang Zhang, Lishan Liu, Yonghong Wu, Yujun Cui}

\address{Xinguang Zhang (corresponding author)\newline
School of Mathematical  and Informational Sciences,
Yantai University, Yantai 264005, Shandong, China. \newline
Department of Mathematics and Statistics,
Curtin University of Technology,
Perth,  WA 6845, Australia}
\email{xinguang.zhang@curtin.edu.au}

\address{Lishan Liu \newline
School of Mathematical Sciences,
Qufu Normal University,
Qufu 273165, Shandong, China. \newline
Department of Mathematics and Statistics,
Curtin University of Technology,
Perth,  WA 6845, Australia}
\email{mathlls@163.com}

\address{Yonghong Wu \newline
Department of Mathematics and Statistics,
Curtin University of Technology,
Perth,  WA 6845, Australia}
\email{y.wu@curtin.edu.au}

\address{Yujun Cui \newline
Department of Mathematics,
Shandong University of Science and Technology,
Qingdao, 266590, Shandong, China}
\email{cyj720201@163.com}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted March 8, 2018. Published July 31, 2018.}
\subjclass[2010]{35J50, 35J92}
\keywords{Modified nonlinear Schr\"odinger equation; dual approach;
\hfill\break\indent critical point theorem; multiplicity; variational methods}

\begin{abstract}
 In this article, we focus on the existence of infinitely many weak solutions
 for the modified nonlinear Schr\"odinger equation
 $$
 -\Delta u+V(x) u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)
 ^{\frac{2-\alpha}2}}=f(x,u),\quad   \text{in } \mathbb{R}^N,
 $$
 where $1\leq\alpha<2$, $f \in C( \mathbb{R}^N \times \mathbb{R}, \mathbb{R})$.
 By using a symmetric mountain pass theorem  and dual approach, we prove
 that the above equation has  infinitely many high energy solutions.
\end{abstract}

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

\section{Introduction}

The quasilinear Schr\"odinger equation
\begin{equation}
-\Delta u+V(x)u-k\Delta(u^2)u=f(x,u), \quad x\in \mathbb{R}^N \label{e1.1}
\end{equation}
is referred as a modified form of the  nonlinear Schr\"odinger equation
\begin{equation}
iz_t+\Delta z-\omega(x)z+\kappa \Delta(h(|z|^2))h'(|z|^2)z+g(x,z) =0, \quad
x\in \mathbb{R}^{n},\label{e1.2}
\end{equation}
where $\omega$ is a given potential, $h$ and $g$ are real functions and $\kappa$
is a real constant.  \eqref{e1.1} is related to the existence of standing waves
solutions of \eqref{e1.2}. In fact, let $z(t,x)=e^{-i\beta t}u(x)$,
by exploring the Lorentz invariance equation \eqref{e1.2}, we can get a solitary
traveling wave and a corresponding equation of elliptic type which has a
formal variational structure like \eqref{e1.1}  for suitable $\omega,h$ and $g$.

 Many researchers focus on the nonlinear Schr\"odinger equation \eqref{e1.2} because
it can model many important physical phenomena \cite{1,Laedke,Nakamura,Porkolab}.
 If $h(s)=s$, it describes the time evolution of the condensate
wave function in superfluid film for plasma physics in Kurihara \cite{1}, and if
$h(s)=(1+s)^{1/2}$,   the equation \eqref{e1.2} models the self-channeling
of a high-power ultrashort laser in matter  \cite{2} and
 the  Heidelberg ferromagnetism \cite{3}.


Because of the strong physical background, \eqref{e1.1} has attracted a lot of
attention from mathematics science field.
In the case of $k=0$, by using the mountain pass theorem (for the impact of the
mountain pass theory in nonlinear analysis, we refer reader to see Pucci and
Radulescu \cite{A1},  Ghergu and Radulescu  \cite{A3}), Bahrouni et al \cite{A7}
 established infinitely many solutions for the following nonlinear
Schr\"odinger equation
\begin{equation*}
-\Delta u+V(x)u=a(x)g(u), \; x\in \mathbb{R}^N (N\ge3)\label{Bahrouni},
\end{equation*}
where $V$ and $a$ are functions changing sign and the nonlinearity $g$ has
a sublinear growth. Recently,
a Schr\"odinger-Maxwell system involving sublinear terms was studied in 
\cite{A6}, and  the existence of at least two non-trivial solutions as 
well as the stability of system   was established via to a recent 
Ricceri-type result.
In addition, for the radial case of Schr\"odinger equations and systems,
 many excellent works have been reported, we refer the reader to
\cite{A7,A2,A4,A5,A10,A9,A8,Z66}.
However, if the  Schr\"odinger equation contains a quasilinear and non-convex
diffusion term $\Delta(u^2)u$,   some unpredictable
difficulties will appear, such as no suitable space where the energy 
functional is well defined or the functional is not  $C^1$-class 
except for $N = 1$ (see \cite{12}).  In order to overcome these
 difficulties, Liu, Wang and Wang \cite{13}  (see also \cite{14}) 
introduced a technique of changing variables, i.e., dual approach  
to rewrite the energy functional with new variable and to find solutions 
of an auxiliary semilinear equation.  Following this technique, 
many good results on various modified forms \eqref{e1.1} of \eqref{e1.2} 
have been reported, see \cite{5,11, 15, 7,zhang3,zhang2,20189, 9,10,6}. 
Recently, Cheng and Yang \cite{16} studied the model of self-channeling 
of a high-power ultrashort laser in matter which has form of
 a nonlinear Schr\"odinger equation
\begin{equation}
\begin{gathered}
-\Delta u+K u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{\frac{2-\alpha}2}}
=|u|^{q-1}u+|u|^{p-1}u,\\
u\in H^1(\mathbb{R}^N),\quad  K>0,\quad N\ge3, \quad \alpha\ge1,\quad
 2<q+1<p+1<\alpha 2^*,
\end{gathered} \label{5}
 \end{equation}
by using a change of variables and Mountain pass theorem,
 the nontrivial solution of the equation \eqref{5} has been established.
However, we notice that the potential $V(x)=K$ is bounded, and the infinitely
many solutions with high energy have not been studied for a more general
nonlinear term.
Thus motivated by the above work, in this paper, we are concerned with
the existence of infinitely many high energy solutions for the
 quasilinear Schr\"odinger equation
\begin{equation}
-\Delta u+V(x) u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2
)^{\frac{2-\alpha}2}}=f(x,u),\quad  \text{in } \mathbb{R}^N,\label{4}
\end{equation}
where $1\le\alpha<2$,  $f \in C( {\mathbb{R}}^N \times \mathbb{R}, \mathbb{R})$ and
$V(x)$ satisfies
\begin{itemize}
\item[(A1)]  $V\in C(\mathbb{R}^N,\mathbb{R})$, $V_0 := \inf V(x) > 0$
 and for every $\Lambda> 0$
$$
\operatorname{meas}(\{x\in \mathbb{R}^N:V(x) <\Lambda \})
< +\infty,
$$
where meas denotes Lebesgue measure in $\mathbb{R}^N$.
\end{itemize}
Our research is also closely related to some work by Sun et al
\cite{24,23,29,25,26}, Mao et al \cite{27, shen,30,A11}, Liu et al \cite{28},
Elisandra \cite{Elisandra}, Shao\cite{A12}, Shi and Chen \cite{hongxia3},
Zhang et al \cite{H68,A14,A13} 
and variational methods for ordinary differential
equations \cite{A16,A20,A19,A15,A17,A18} and partial differential
equations \cite{X67,A21,A22,A27,A24,A25,A26,A23},  where the authors
obtained some interesting theoretical results.

At the end of this section, we state a version of symmetric mountain pass
theorem due to Rabinowize \cite{17}, the proof of our main result will depend on it.

\begin{lemma} \label{lem1.1}
Let $E$ be an infinite dimensional Banach space and let
$I\in C^1(E, \mathbb{R})$ be even, satisfy (PS)-condition, and $I(0) = 0$. If
$E = V \oplus X$, where $V$ is finite dimensional  and $I$ satisfies
\begin{itemize}
\item[(i)] there are constants  $ \rho, \delta > 0$ such that
$I|_{\partial B_\rho\cap X} \ge \delta$, and

\item[(ii)] for each finite-dimensional subspace $E'\subset E$, there is an
$R = R(E')$ such that $I |_{E'\backslash B_R} \le0$.
\end{itemize}
 Then $I$  possesses an unbounded sequence of critical values.
\end{lemma}



\section{Variational setting and main results}

The following notation will be adopted in this article.
$L^s({\mathbb{R}^N})$ denotes the usual Lebesgue space with norm
$$
\|u\|_s=\Big(\int_{{\mathbb{R}^N}}|u|^sdx\Big)^{1/s}, \quad 1\le s<\infty.
$$
Let
$$
H^1({\mathbb{R}^N})=\{u\in L^2(\mathbb{R}^N): \nabla u\in L^2(\mathbb{R}^N)\}
$$
with the norm and inner product, respectively,
$$
\|u\|_{H^1}=\Big[\int_{{\mathbb{R}^N}}|\nabla u|^2+u^2dx\Big]^{1/2},\quad
 \langle u,v\rangle_{H^1}=\int_{{\mathbb{R}^N}}(\nabla u \cdot \nabla v+uv)dx.
$$
Now under the assumption (A1), we define our work space
$$
E=\big\{u\in H^1({\mathbb{R}^N}):\int_{\mathbb{R}^N} (|\nabla u|^2+V(x) u^2)dx
<+\infty\big\}
$$
with the norm and inner product, respectively,
$$
\|u\|=\Big[\int_{{\mathbb{R}^N}}|\nabla u|^2+V(x)u^2dx\Big]^{1/2},\quad
\langle u,v\rangle=\int_{{\mathbb{R}^N}}(\nabla u \cdot \nabla v+
v(x)u v)dx.
$$
 It is well known that if the assumption (A1) holds, then the embedding
$E\hookrightarrow L^s({\mathbb{R}^N})$ is continuous for $s\in [2,2^*]$
and there exists a constant $c_s>0$, $2\le s\le2^*$ such that
 $$
\|u\|_s\le c_s \|u\|,\quad \forall u\in E.
$$
In addition, from \cite{19,20}, we have the following compactness lemma.

 \begin{lemma} \label{lem2.1}
Under assumption {\rm (A1)}, the embedding $E\hookrightarrow L^s({\mathbb{R}^N})$
is compact  for $s\in [2,2^*)$.
\end{lemma}

Normally,  the solutions of  \eqref{4}  are the critical points of the
 functional
 $$
J(u)=\frac12\int_{\mathbb{R}^N} \big[1+\frac{\alpha^2u^2}{2(1+u^2)^{2-\alpha}} \big]
|\nabla u|^2dx+\frac12\int_{\mathbb{R}^N} V(x) u^2dx
-\int_{{\mathbb{R}^N}}F(x,u)dx,
$$
 where $F(x,s)=\int_0^s f(x,\xi)d\xi$.  But the natural associated functional
 $J(u)$  may not be well defined and  is not G\^{a}teaux differentiable
functional in the corresponding Sobolev space $E$. To avoid these obstacles,
 we introduce a new function so that the dual approach can be used for establishing
our results.
Let
$$
g(t)=\sqrt{1+\frac{\alpha^2 t^2}{2(1+t^2)^{2-\alpha}}}
$$
and make a change of variable
$$
v=G(u)=\int_0^u g(t)dt.
$$
Clearly, $g(t)$ is monotonous on $|t|$, which implies that the inverse
function $G^{-1}(t)$ of $G(t)$ exists, thus similar to \cite{16}, we have
an equivalent functional for the natural associated functional $J(u)$
 \begin{equation}
I(v)=\frac12\int_{\mathbb{R}^N} |\nabla v|^2dx
+\frac12\int_{\mathbb{R}^N} V(x) |G^{-1}(v)|^2dx
-\int_{{\mathbb{R}^N}}F(x,G^{-1}(v))dx.
\end{equation}
Based on the properties of $G^{-1}(v)$ (see Lemma \ref{lem2.2}  below),
$I(\cdot)$ is well defined on $E$ and $I(v)\in C^1(E,\mathbb{R})$ if and only if
 $\int_{{\mathbb{R}^N}}F(x,G^{-1}(\cdot))dx$ has the same property as the
functional $I$, i.e., if $\int_{{\mathbb{R}^N}}F(x,G^{-1}(\cdot))dx$ is
continuously differential on $E$, then $I(v)\in C^1(E,\mathbb{R})$, and for
any $w\in C_0^\infty (\mathbb{R})$, we have
$$
\langle I'(v),w \rangle=\int_{\mathbb{R}^N} \nabla v \nabla w dx
+\int_{\mathbb{R}^N} V(x) \frac{G^{-1}(v)}{g\left(G^{-1}(v)\right)}{w} dx
-\int_{{\mathbb{R}^N}}\frac{f\left(x,G^{-1}(v)\right)}{g\left(G^{-1}(v)\right)}{w} dx.$$
The  critical points of $I$ are then weak solutions of the semilinear Schr\"odinger  equation
\begin{equation}
 -\Delta v=-V(x) \frac{G^{-1}(v)}{g\big(G^{-1}(v)\big)}
+\frac{f\big(x,G^{-1}(v)\big)}{g\big(G^{-1}(v)\big)},\quad
 x\in {\mathbb{R}^N}.\label{2.1}
\end{equation}
Thus  to obtain the existence of the weak solutions for the quasilinear
Schr\"odinger  equation \eqref{4}, it is sufficient to study the existence
of the weak solutions for the equivalent form \eqref{2.1} of \eqref{4}.

In this article, we assume that the nonlinearity $f$ in problem \eqref{4}
satisfies the following assumptions:
\begin{itemize}
\item[(A2)] $f(x,-t)=-f(x,t)$ for all $(x,t)\in\mathbb{R}^N\times \mathbb{R}$.

\item[(A3)] there exists $c>0$ such that  $|f(x,t)|\le c (1+|t|^{r-1})$ for some
$2\alpha<r<2^*\alpha$, where $2^*=\frac{2N}{N-2}$ if $N\ge3$ and
$2^*=\infty$  if $N=2$.

\item[(A4)] $f(x,t)=o(|t|)$ uniformly in $x$ as $|t|\to 0$.

\item[(A5)] $\lim_{|t|\to \infty}\frac{|F(x,t)|}{|t|^{2\alpha}}=+\infty$
uniformly for $x\in\mathbb{R}^N$, where $F(x,t)=\int_0^t f(x,s)ds$.

\item[(A6)] there exists a constant $\mu>2\alpha$ such that
$$
f(x,t)G(t)-\mu F(x,t)g(t)\ge0,
$$
for  all $(x,t)\in\mathbb{R}^N\times\mathbb{R}$.
\end{itemize}


On the existence of  infinitely many  high energy  solutions  we have the
following result.


\begin{theorem} \label{thm2.1}
Suppose that  {\rm (A1)--(A6)} are satisfied. Then  the  \eqref{4}
admits  a sequence of weak solutions $\{u_n\}\subset E$ such that
$\|u_n\|\to \infty$ and $J(u_n)\to \infty$ as $n\to \infty$.
\end{theorem}

To prove our main result, some properties of $G^{-1}(t)$ will be introduced
so that we can discuss the geometric structure of $I$ more conveniently.

 \begin{lemma} \label{lem2.2}
$g(t)$ and $G^{-1}(t)$  satisfy the following properties:
\begin{itemize}
\item[(G1)]  $g(t)\ge1, \forall t \in \mathbb{R}$;

\item[(G2)] $|G^{-1}(t)|\le |t|, \forall t \in  \mathbb{R};$

\item[(G3)] $\lim_{t\to0}\frac{G^{-1}(t)}{t}=1;$

\item[(G4)] if $\alpha>1$, then  $\lim_{t\to\infty}\frac{|G^{-1}(t)|^\alpha}{t}=\sqrt{2};$  if  $\alpha=1$,
then $\lim_{t\to\infty}\frac{|G^{-1}(t)|}{t}=\sqrt{\frac23};$

\item[(G5)] there exist a positive constant such that
\[
|G^{-1}(t)|\ge\begin{cases}
 C|t|,& |t|\le 1,\\
 C t^{1/\alpha},& |t|\ge 1;
\end{cases}
\]

\item[(G6)]
$$
\frac{G^{-1}(t)t}{g(G^{-1}(t))}\le (G^{-1}(t))^2,\quad  \forall t \in  \mathbb{R}.
$$

\item[(G7)] $|G(t)|\le g(t)|t|$, for any $t\in \mathbb{R}$;

\item[(G8)] for any $t\in\mathbb{R}$, we have $\frac{tg'(t)}{g(t)}\le T(\alpha)$,
 where
$$
T(\alpha)= \begin{cases}
  \alpha-1, &  \alpha\ge\alpha_1\approx 1.1586,\\
\frac{\alpha^2}{2}\big(\frac{3-\alpha}{2-\alpha}\big)^{\alpha-3},
& 1\le \alpha<\alpha_1,
\end{cases}
$$
 especially, for the case $1\le \alpha<\alpha_1$, for accuracy, $T(\alpha)$
can be taken as  $\rho(s_0)$, where $s_0$ satisfies $\rho'(s_0)=0$
and $ 1\le \alpha<\alpha_1$,
$$
\rho(s)=\frac{(\alpha-1)\alpha^2 s(1+s)^{\alpha}
+(2-\alpha)\alpha^2s(1-s)^{\alpha-1}}{2(1+s^2)+\alpha^2 s(1+s)^\alpha},\quad  s\ge0.
$$

\item[(G9)] for each $\lambda>1$, one has
$$
|{G^{-1}(\lambda t)}|^2\le \lambda^2|G^{-1}(t)|^2,\quad
 \forall t \in  \mathbb{R}.
$$

\item[(G10)] the function $(G^{-1}(t))^2$ is strictly convex, and especially
$$
|{G^{-1}(\lambda t)}|^2\le \lambda|G^{-1}(t)|^2,\quad  \forall t \in  \mathbb{R},
\; \lambda\in[0,1].
$$

\item[(G11)] there exists a constant $c>0$ such that
$|G^{-1}(t)|^\alpha\le c|t|$ for all $t \in  \mathbb{R}$.
\end{itemize}
\end{lemma}

\begin{proof}
The proof of (G2)--(G4) and (G8) can be found in \cite{16}.
By the definition of $g$ and direct calculation,  (G1) and (G6) hold.
In addition, since $G^{-1}$ is an odd function, (G5) and (G11) are consequences of
(G3) and (G4), moreover (G7) is also consequence of \cite{16,14}.
To prove (G9), by (G7), we have
\[
t=G(G^{-1}(t))\le g(G^{-1}(t))G^{-1}(t), \quad \text{for all } t\ge0.
\]
Thus
\[
\frac{[(G^{-1}(s))^2]'t}{(G^{-1}(t))^2}
=\frac{2G^{-1}(t)(G^{-1}(t))'t}{(G^{-1}(t))^2}
=\frac{2G^{-1}(t)t}{g(G^{-1}(t))(G^{-1}(t))^2}
\le \frac{2(G^{-1}(t))^2}{(G^{-1}(t))^2}=2,
\]
for all $t\ge0$. Then
\[
\ln\Big(\frac{(G^{-1}(\lambda t))^2}{(G^{-1}(t))^2}\Big)
= \int_t^{\lambda t}\frac{[(G^{-1}(s))^2]'}{(G^{-1}(s))^2}ds
 \le 2\ln \lambda=\ln \lambda^2,
\]
for all $t\ge0$ and $\lambda>1$, which implies that
$$
(G^{-1}(\lambda t))^2 \le \lambda^2{(G^{-1}(t))^2},\quad \text{for all }
t\ge0.
$$
Since $G^{-1}$ is an odd function, and $(G^{-1})^2$ is a even function,
so the above inequality holds for all $t\in \mathbb{R}$.

 In the end,  we prove (G10). In fact, for $1\le\alpha<\alpha_1$, we have that
$\phi(\alpha)=\frac{\alpha^2}{2}\big(\frac{3-\alpha}{2-\alpha}\big)^{\alpha-3}$
is increasing and $0<\phi(\alpha)<1$, thus  by $(g_8)$, for any $1\le\alpha<2$ 
and $s\in \mathbb{R}$, we have
\begin{equation*}
\frac{sg'(s)}{g(s)}\le T(\alpha)<1,
\end{equation*}
 which yields
$$
[(G^{-1}(s))^2]''=\frac2{g^2(G^{-1}(t))}-\frac{2G^{-1}(t)
g'(G^{-1}(t))}{g^3(G^{-1}(t))}>0.
$$
And then, from the convexity of $(G^{-1}(t))^2$, for all $\lambda\in[0,1]$,
one gets
$$
|{G^{-1}(\lambda t)}|^2\le \lambda|G^{-1}(t)|^2,\quad
 \forall t \in  \mathbb{R}.
$$
\end{proof}


\begin{lemma} \label{lem2.3}
Assume that $\{v_n\}\subset E$ is a (PS)-sequence of $I$.
Then $\{v_n\}$ is bounded in $E$.
\end{lemma}

\begin{proof}
Suppose  $\{v_n\}\subset E$ is a (PS)-sequence of $I$, that is
\begin{equation} \label{e2.3}
I(v_n)\to c, \quad (1+\|v_n\|)I'(v_n)\to 0, \quad \text{as } n\to\infty.
\end{equation}
By using \eqref{e2.3}, (A1), (G6) and (A6), we get
\begin{equation}
\begin{aligned}
 c+o(1)
&=I(v_n)-\frac1{\mu}\langle I'(v_n),v_n\rangle\\
&=\Big(\frac12-\frac1\mu\Big)\int_{\mathbb{R}^N}|\nabla v_n|^2dx
 +\frac12 \int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx \\
&\quad -\frac1{\mu}\int_{\mathbb{R}^N} V(x)\frac{G^{-1}(v_n)v_n}{g(G^{-1}(v_n))}dx
  +\frac1{\mu}\int_{\mathbb{R}^N}\frac{f(x,G^{-1}(v_n))v_n}{g(G^{-1}(v_n))}dx \\
&\quad -\int_{\mathbb{R}^N} F(x,G^{-1}(v_n))dx\\
&\ge \Big(\frac12-\frac1\mu\Big)\int_{\mathbb{R}^N}|\nabla v_n|^2dx
 +\Big(\frac12-\frac1\mu\Big)\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx\\
&\quad +\frac1{\mu}\int_{\mathbb{R}^N}
 \Big(\frac{f(x,G^{-1}(v_n))v_n}{g(G^{-1}(v_n))}-\mu F(x,G^{-1}(v_n))\Big)dx\\
&\ge \Big(\frac12-\frac1\mu\Big)\int_{\mathbb{R}^N}|\nabla v_n|^2dx
 +\Big(\frac12-\frac1\mu\Big)\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx,
 \end{aligned}
\end{equation}
which implies that there exists  a constant $C_1>0$ such that
\begin{equation} \label{e2.5}
 \int_{\mathbb{R}^N}|\nabla v_n|^2dx+\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx
\le C_1.
\end{equation}
Obviously, from \eqref{e2.5}, if there exists a constant $C_2>0$ such that
 \begin{equation}\label{e2.6}
  \int_{\mathbb{R}^N}|\nabla v_n|^2dx+\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx
\ge C_2 \|v_n\|^2,
\end{equation}
 then $\{v_n\}$ is bounded in $E$. To do this, let
\begin{equation}\label{e2.7}
 \|v_n\|_0^2=:\int_{\mathbb{R}^N}|\nabla v_n|^2dx
+\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx,
\end{equation}
and $v_n\neq 0$ (if $v_n=0$, the conclusion obviously holds).
Suppose \eqref{e2.6} is not true, then passing to a subsequence, one has
$$
\lim_{n\to +\infty}\frac{\|v_n\|_0^2}{\|v_n\|^2}=0.
$$
Set
$$
u_n=\frac{v_n}{\|v_n\|}, \quad
 k_n=\frac{(G^{-1}(v_n))^2}{\|v_n\|^2},
$$
then we have
\begin{equation}
\int_{\mathbb{R}^N}|\nabla u_n|^2dx+\int_{\mathbb{R}^N} V(x)k_n(x)dx\to 0,\quad
 \ n\to \infty.
\end{equation}
Thus
\begin{equation} \label{e2.9}
\int_{\mathbb{R}^N} |\nabla u_n|^2 dx \to 0,\quad
\int_{\mathbb{R}^N} V(x)k_n(x) dx \to 0,\quad
\int_{\mathbb{R}^N} V(x) u^2_n dx \to1.
\end{equation}
Now according to the strategy in \cite{15,9}, we claim that for each
$\varepsilon>0$, there exists a constant $C_2>0$ such that
$\operatorname{meas}(B_n)\le \varepsilon$, where $\operatorname{meas}(\cdot)$
denotes the standard Lebesgue measure and
$$
B_n=\{x\in {\mathbb{R}^N}: |v_n|\ge C_2 \}.
$$
Otherwise,  there exists $\varepsilon_0>0$ and a subsequence of $\{v_n\}$
(still  denoted by $\{v_n\}$) such that for any positive integer $n$
 $$
\operatorname{meas}(A_n)\ge \varepsilon_0,
$$
where $A_n=\{x\in {\mathbb{R}^N}: |v_n|\ge n \}$.
By (G5)  and $(V_0)$, we have
$$
\|v_n\|_0^2\ge\int_{\mathbb{R}^N} V(x)|G^{-1}( v_n)|^2dx
\ge \int_{A_n} V(x)|G^{-1}( v_n)|^2dx
\ge C_3n^{1/\alpha}\varepsilon_0\to +\infty,
$$
as $n\to \infty$,
 which contradicts with \eqref{e2.5}, thus our claim is true.

Next notice that if $v_n\in {\mathbb{R}^N} \setminus B_n$,
it follows from (G5), (G9) and (G10) that
$$
\frac{C}{C_2^2}v_n^2\le \Big(G^{-1}\big( \frac{v_n}{C_2}\big)\Big)^2
\le C_3\Big(G^{-1}(v_n)\Big)^2,
$$
which implies
\begin{equation} \label{e2.10}
\begin{aligned}
\int_{{\mathbb{R}^N} \setminus B_n} V(x) u^2_n dx
&\le C_4 \int_{{\mathbb{R}^N} \setminus B_n} V(x)
 \frac{\big(G^{-1}(v_n)\big)^2}{\|v_n\|} dx
\le C_4\int_{{\mathbb{R}^N}} V(x)k_n(x)dx \\
&\to 0,\quad \text{as } n\to \infty.
\end{aligned}
\end{equation}
On the other hand, by the absolute equicontinuity of integral \cite{22},
there exists $\varepsilon>0$ such that whenever $\Omega\subset \mathbb{R}^N$
and $\operatorname{meas}(\Omega)<\varepsilon$
\begin{equation} \label{e2.11}
\int_{\Omega} V(x) u^2_n dx\le \frac12.
\end{equation}
Thus it follows from  \eqref{e2.10} and \eqref{e2.11} that
\[
\int_{{\mathbb{R}^N}} V(x) u^2_n dx
=\int_{B_n} V(x) u^2_n dx+\int_{{\mathbb{R}^N} \setminus B_n} V(x) u^2_n dx
\le \frac12+o(1),
\]
 which implies that $1\le \frac12$, a contradiction.
Thus \eqref{e2.6} is indeed true, and then $\{v_n\}$ is bounded in $E$.
\end{proof}


\begin{lemma} \label{lem2.4}
Assume that $\{v_n\}$ is bounded in  $E$, then for any $v\in E$, there exists
a constant $C_5>0$ such that
\begin{equation} \label{e2.12}
\begin{aligned}
&\int_{\mathbb{R}^N}|\nabla(v_n-v)|^2dx+\int_{\mathbb{R}^N}
 V(x)[\frac{G^{-1}(v_n)}{g(G^{-1}(v_n))}
 -\frac{G^{-1}(v))}{g\left(G^{-1}(v)\right)}](v_n-v)dx \\
&\ge C_5\|v_n-v\|^2.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Let $v_n\neq v$, otherwise, the conclusion is trivial. Set
\begin{equation} \label{e2.13}
w_n=\frac{v_n-v}{\|v_n-v\|},\quad
h_n(x)=\left(\frac{G^{-1}(v_n(x))}{g(G^{-1}(v_n(x)))}
-\frac{G^{-1}(v(x)))}{g\left(G^{-1}(v(x))\right)}\right)
\big/ (v_n(x)-v(x)).
\end{equation}
To obtain \eqref{e2.12}, it suffices to prove that there exists a constant $C_5>0$
such that
\begin{equation} \label{e2.14}
\int_{\mathbb{R}^N}|\nabla w_n|^2dx+\int_{\mathbb{R}^N} V(x)h_n(x)w^2_ndx
\ge C_5.
\end{equation}
To do this, we argue it by contradiction. Assume that
 \begin{equation} \label{e2.15}
\int_{\mathbb{R}^N}|\nabla w_n|^2dx+\int_{\mathbb{R}^N} V(x)h_n(x)w^2_ndx\to 0.
\end{equation}
By (G8),
 \begin{equation} \label{e2.16}
\frac{d}{dt}[\frac{G^{-1}(t)}{g(G^{-1}(t))}]
=\frac1{g^2(G^{-1}(t))}-\frac{G^{-1}(t)g'(G^{-1}(t))}{g^3(G^{-1}(t))}>0,
\end{equation}
which implies that $\frac{G^{-1}(t)}{g(G^{-1}(t))}$ is strictly increasing
and for each $C_6>0$ there exists a constant $\delta_1>0$ such that
\begin{equation}
\frac{d}{dt}\Big[\frac{G^{-1}(t)}{g(G^{-1}(t))}\Big]\ge \delta_1
\end{equation}
as $|t|\le C_6$. Moreover,  by the Mean Value Theorem and \eqref{e2.16},
the second equality of \eqref{e2.13} becomes
\begin{equation} \label{e2.18}
\begin{aligned}
h_n(x)&=\Big(\frac{G^{-1}(v_n(x))}{g(G^{-1}(v_n(x)))}
-\frac{G^{-1}(v(x))} {g\left(G^{-1}(v(x))\right)} \Big)\big/ (v_n(x)-v(x))\\
& =\frac{d}{dt}[\frac{G^{-1}(t)}{g(G^{-1}(t))}]{\Big|}_{t=v_n(x)
 +\theta (v_n(x)+v(x))}\ge0.
\end{aligned}
\end{equation}
It follows from \eqref{e2.13}, \eqref{e2.14} and \eqref{e2.18} that
\begin{equation} \label{e2.19}
 \int_{\mathbb{R}^N} |\nabla w_n|^2 dx \to 0,\quad
  \int_{\mathbb{R}^N} V(x)h_n(x) w^2_n dx \to 0,\quad
\int_{\mathbb{R}^N} V(x) w^2_n dx \to1.
\end{equation}
Thus similar to the argument of \eqref{e2.10} and \eqref{e2.11}, one can
get a contradiction. So \eqref{e2.12} holds.
\end{proof}

Now let $\{e_i\}$  be an orthonormal basis of $E$ and define $X_i=\mathbb{R}e_i$,
then $E =\oplus_{i=1}^{\infty} X_i$. Let
$$
V_n=\oplus_{i=1}^{n} X_i,\quad
W_n= {\oplus_{i=n}^{\infty} X_i},\quad  n\in \mathbb{Z},
$$
then $V_n$ is finite dimensional.
By \cite[Lemma 3.8]{21}, we have the following conclusion.

\begin{lemma} \label{lem2.5}
Assume $(V_0)$ and $2\le s<2^*$, then
$\sup_{v\in W_n,\|v=1\|}\|v\|_s\to 0$ as $n\to \infty$.
\end{lemma}


\begin{lemma} \label{lem2.6}
Assume {\rm (A1), (A3)} and {\rm (A4)} hold. Then there exist constants
$\rho,\delta>0$ and positive integer $k\ge1$ such that
$I|_{\partial S_\rho\cap W_k}\ge \delta$ and
$I(v)$ satisfies the (PS)-condition.
\end{lemma}

\begin{proof}
 Firstly, we prove that,  for any $v\in S_\rho$, there exists a positive
constant $C_7$ such that
 \begin{equation} \label{e2.20}
\|v\|_0^2=:\int_{\mathbb{R}^N}|\nabla v|^2dx
+\int_{\mathbb{R}^N} V(x)|G^{-1}( v)|^2dx\ge C_7 \|v\|^2.
\end{equation}
Otherwise, there exists a sequence $\{v_n\}\subset S_\rho$ such that
\[
  \int_{\mathbb{R}^N}|\nabla v|^2dx+\int_{\mathbb{R}^N} V(x)|G^{-1}( v)|^2dx
\le \frac1n \|v\|^2,
\]
which yields
$\|v\|_0^2/\|v\|^2 \to 0$ as $n\to +\infty$. Similar to \eqref{e2.6},
one gets a contraction. Thus  \eqref{e2.20} holds.

On the other hand, for any $\epsilon>0$,  form (A3) and (A4) there exists
${C}_\epsilon>0$ such that
\begin{equation}
|f(x,t)|\le \epsilon |t|+{C}_\epsilon |t|^{r-1}, \quad
 (x,t)\in {\mathbb{R}^N}\times \mathbb{R}.
\end{equation}
Moreover it follows from Lemma \ref{lem2.5} and $2\alpha<r<2^*\alpha$ that there exists
an integer $k\ge1$ such that
\begin{equation} \label{e2.22}
\|v\|^2_2\le C_8 \|v\|^2, \quad
\|v\|^{\frac r\alpha}_{\frac r\alpha}\le C_9 \|v\|^{\frac r\alpha},\quad
 \forall v\in W_k.
\end{equation}
 Thus for any $ v\in W_k$ and $v\in S_\rho$, by \eqref{e2.20}-\eqref{e2.22},
(G2) and (G11), we have
\begin{equation} \label{e2.23}
\begin{aligned}
 I(v)&\ge\frac{C_7}2\|v\|^2-\epsilon\int_{{\mathbb{R}^N}}|G^{-1}( v)|^2dx
 -C_\epsilon\int_{\mathbb{R}^N} |G^{-1}( v)|^{r}dx\\
 &\ge\frac{C_7}2\|v\|^2-\epsilon\int_{{\mathbb{R}^N}}|v|^2dx
 -\widetilde{C}_\epsilon\int_{\mathbb{R}^N} |v|^{\frac r \alpha}dx\\
&\ge \frac{C_7}2\rho^2-C_8\epsilon \rho^2- \widetilde{C}_\epsilon
 C_9 \rho^{\frac r\alpha}£º=\delta>0,
\end{aligned}
\end{equation}
for small $\epsilon>0$ and $\rho>0$, that is
$I|_{\partial S_\rho\cap W_k}\ge \delta$.

Let $\{v_n\}\subset E$ be any (PS)-sequence of $I(v)$, i.e; there exists
 $c>0$ such that $|I(v_k)|\le c$ and $I'(v_k)\to 0$ as $k\to \infty$.
From Lemma \ref{lem2.3}, we know $\{v_n\}$ is bounded in  $E$. Thus, up to a
subsequence, we have $v_n\rightharpoonup v$ in $E$.
Moreover, the compactness of embedding  $E\hookrightarrow L^s({\mathbb{R}^N})$
($s\in [2,2^*)$) implies that  $v_n\rightarrow v$ in $L^s({\mathbb{R}^N})$
for any  $2\le s<2^*$ and $v_n(x)\to v(x)$  a.e.\ on ${\mathbb{R}^N}$.

According to (G11) and (G5), we have
\begin{equation*}
|G^{-1}(t)|^\alpha\le c|t| \le cg(G^{-1}(t))|G^{-1}(t)|,\end{equation*}
which yields
\begin{equation}\label{e2.24}
\frac{ |G^{-1}(t)|^{\alpha-1}}{g(G^{-1}(t))}\le c.
\end{equation}
Thus by (G1), (G2), (G11) and \eqref{e2.24}, one has
\begin{equation}
\begin{aligned}
&\Big|\int_{\mathbb{R}^N}\Big[\frac{f(x,G^{-1}(v_n))}{g(G^{-1}(v_n))}
 -\frac{f\left(x,G^{-1}(v)\right)}{g\left(G^{-1}(v)\right)}\Big](v_n-v)dx\Big|\\
&\le \int_{\mathbb{R}^N} \Big[|\frac{f(x,G^{-1}(v_n))}{g(G^{-1}(v_n))}|
 +|\frac{f(x,G^{-1}(v))}{g(G^{-1}(v))}|\Big]|v_n-v|dx\\
& \le\int_{\mathbb{R}^N}[\epsilon\Big(|G^{-1}(v_n)|+|G^{-1}(v)|\Big)
 +{C}_\epsilon\Big(\frac{|G^{-1}(v_n)|^{r-1}}{g(G^{-1}(v_n))}
 +\frac{|G^{-1}(v)|^{r-1}}{g(G^{-1}(v))}\Big)]|v_n-v|dx\\
& \le\int_{\mathbb{R}^N}[\epsilon\left(|v_n|+|v|\right)
 +\widetilde{{C}}_\epsilon\left(|v_n|^{\frac{r-\alpha}{\alpha}}
 +|v|^{\frac{r-\alpha}{\alpha}}\right)]|v_n-v|dx\\
&\le C_{10}(\|v_n\|_{2}+\|v\|_{2})\|v_n-v\|_{2}+C_{11}
 \left(\|v_n\|_{\frac{r}{\alpha}}^{\frac{r-\alpha}{\alpha}}
 +\|v\|_{\frac{r}{\alpha}}^{\frac{r-\alpha}{\alpha}}\right)
 \|v_n-v\|_{\frac{r}{\alpha}}\\& =o_n(1).
\end{aligned} \label{e2.25}
\end{equation}
Thus Lemma \ref{lem2.4}, \eqref{e2.25} and $I'(v_n)\to 0$ imply
\begin{align*}
o(1)&=\langle I'(v_n)-I'(v),v_n-v\rangle\\
&=\int_{\mathbb{R}^N}|\nabla(v_n-v)|^2dx
 +\int_{\mathbb{R}^N} V(x)[\frac{G^{-1}(v_n)}{g(G^{-1}(v_n))}
 -\frac{G^{-1}(v))}{g\left(G^{-1}(v)\right)}](v_n-v)dx\\
&\quad -\int_{\mathbb{R}^N}[\frac{f(x,G^{-1}(v_n))}{g(G^{-1}(v_n))}
 -\frac{f\left(x,G^{-1}(v)\right)}{g\left(G^{-1}(v)\right)}](v_n-v)dx\\
&\ge C_5\|v_n-v\|^2+o_n(1),
\end{align*} %\label{2.3}
 which yields $v_n\to v$ in $E$.  The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.7}
For  each finite-dimensional subspace $E'\subset E$, there is a constant
$R>\rho$ such that $I|_{E'\setminus B_R}\le0$.
\end{lemma}

\begin{proof}
Suppose that the conclusion of the lemma is not invalid  for some finite-dimensional
subspace $E'\subset E$. Then there is a sequence $\{v_n\}\subset E'$
such that $\|v_n\|\to \infty$ and $I(v_n)>0$, that is
\begin{equation}
\frac12\int_{\mathbb{R}^N} \left(|\nabla v_n|^2+ V(x) |G^{-1}(v_n)|^2\right)dx
>\int_{{\mathbb{R}^N}}F(x,G^{-1}(v_n))dx.
\end{equation}
By (G2), we have
\begin{equation} \label{e2.28}
\int_{{\mathbb{R}^N}}\frac{F(x,G^{-1}(v_n))dx}{\|v_n\|^2}<\frac12.
\end{equation}

On there other hand, let $w_n=\frac{v_n}{\|v_n\|}$.
 Then up to a sequence, we can assume that $w_n \rightharpoonup w$ in $E$,
$w_n \to w$ in $L^s({\mathbb{R}^N}), s\in [2,2^*)$, $w_n \rightharpoonup w$ for a.e.
  $x\in {\mathbb{R}^N}$. Let $\Lambda=\{x\in {\mathbb{R}^N}: w(x)\neq 0\}$ and
$\Lambda_1=\{x\in {\mathbb{R}^N}: w(x)=0\}$, we assert $meas \Lambda=0$.
In fact, if not, by {(A5)}, (G4)  and the Fatou's Lemma, one has
\begin{equation} \label{e2.29}
\int_{\Lambda}\frac{F(x,G^{-1}(v_n))}{\|v_n\|^2}dx
=\int_{\Lambda}\frac{F(x,G^{-1}(v_n))dx}{(G^{-1}(v_n))^{2\alpha}}
\frac{(G^{-1}(v_n))^{2\alpha}}{v_n^2}w_n^2dx\to +\infty.
\end{equation}
On the other hand, by assumptions (A3)--(A5), there exists a constant
$C_{12}>0$ such that
$$
F(x,t)\ge -C_{12}t^2,\quad  \forall (x,t)\in {\mathbb{R}^N}\times \mathbb{R},
$$
which implies that
$$
\int_{\Lambda_1}\frac{F(x,G^{-1}(v_n))}{\|v_n\|^2}dx
\ge -C_{12}\int_{\Lambda_1}\frac{(G^{-1}(v_n))^2}{\|v_n\|^2}dx
\ge-C_{12}\int_{\Lambda_1} w_n^2dx.
$$
Since $w_n\to w$ in $L^2({\mathbb{R}^N})$, by \cite{21}, there exists a function
$h\in L^2({\mathbb{R}^N})$ such that
$$
|w_n(x)|\le h(x),\quad\text{a.e. }  x \in {\mathbb{R}^N}.
$$
Thus Lebesgue's dominated convergence theorem guarantees
\begin{equation} \label{e2.30}
\liminf_{n\to \infty} \int_{\Lambda_1}\frac{F(x,G^{-1}(v_n))}{\|v_n\|^2}dx\ge0.
\end{equation}
Consequently, \eqref{e2.29} and \eqref{e2.30} yield
\begin{equation}
\liminf_{n\to \infty} \int_{{\mathbb{R}^N}}\frac{F(x,G^{-1}(v_n))}{\|v_n\|^2}dx
=+\infty,
\end{equation}
which contradicts  \eqref{e2.28}. So  $\operatorname{meas} \Lambda=0$ and
$w(x)=0$ a.e $x\in {\mathbb{R}^N}$. According the fact of all norms are equivalent
on the finite dimensional space and Sobloev embedding theorem, there is a
constant $d>0$ such that
$$
0=\lim_{n\to \infty}\|w_n\|^2_2\ge \lim_{n\to \infty}d\|w_n\|^2=d,
$$
a contradiction, and the desired conclusion is obtained.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Let $V=V_n,\ X=W_n$, then $E = V \oplus X$, $V$ is a finite-dimensional space.
Obviously, by $(f_1)$, we know the functional $I$ is even and $I(0)=0$.
 Lemma \ref{lem2.6} implies that $I$ satisfies (PS)-condition, and by
Lemmas \ref{lem2.6} and \ref{lem2.7}, $(i)$  and $(ii)$ of 
Lemma \ref{lem1.1} are also satisfied.
Thus, by Lemma \ref{lem1.1}, $I$ possesses a sequence of  critical
points $\{v_n\}\subset E$ such that $I(v_n) \to \infty$  as
$n\to \infty$, i.e., the problem \eqref{4} has a sequence of  solutions
 $\{u_n\}\subset E$  such that
and $\|u_n\|\to \infty$ and $J(u_n)\to \infty$ as $n\to \infty$, where
$u_n = G^{-1}(v_n)$.
\end{proof}

\section{Further results}

 In this section, we obtain infinitely many high energy  solutions for \eqref{4}
by using some  easily verifiable  assumptions:
\begin{itemize}
\item[(A7)] there exists a constant $M>0$ such that
 $$
M_0=\inf_{(x,t)\in\mathbb{R}^N\times [ G^{-1}(M),+\infty)}F(x,t)>0.
$$
\end{itemize}

\begin{theorem}  \label{thm3.1}
Suppose that  {\rm (A2)--(A4), (A6), (A7)}   are satisfied.
Then \eqref{4} admits  a sequence of weak solutions $\{u_n\}\subset E$ such that
$\|u_n\|\to \infty$ and $J(u_n)\to \infty$ as $n\to \infty$.
\end{theorem}

\begin{proof} We only need to prove that the assumption (A7) is stronger than
(A5). To do this, for any $(x,t)\in \mathbb{R}^N\times \mathbb{R}$, let
$$
\varphi(s)=F\Big(x, G^{-1}\big(\frac{t}{s}\big)\Big)s^\mu, \quad s\ge1.
$$
It follows from  (A6) and (A7) that
 \begin{equation} \label{e3.1}
\begin{aligned}
\varphi'(s)&=f\Big(x, G^{-1}\big(\frac{t}{s}\big)\Big)
\big(-\frac{t}{s^2}\big)\big[G^{-1}\big(\frac{t}{s}\big)\big]' s^\mu
 +\mu F\Big(x,G^{-1}\big(\frac{t}{s}\big)\Big)s^{\mu-1}\\
& =\frac{s^{\mu-1}}{g(G^{-1}\big(\frac{t}{s}\big))}
\Big[-\frac{t}{s}f\Big(x, G^{-1}\big(\frac{t}{s}\big)\Big)
+\mu F\Big(x, G^{-1}\big(\frac{t}{s}\big)\Big)
 g\Big(G^{-1}\big(\frac{t}{s}\big)\Big)\Big]\\
&\le0,
\end{aligned}
 \end{equation}
which implies that
 $\varphi(s)$ is decreasing on $[1,+\infty)$. Thus  for $|t|>M$,
notice that $\frac{M|t|}{t}$ is an odd function and $F$ is a even function, we have
 $$
\varphi(1)=F(x,G^{-1}(t))\ge \varphi\big(\frac{|t|}{M}\big)
=F\Big(x,G^{-1}\big(\frac{M|t|}{t}\big)\Big)\big(\frac{|t|}{M}\big)^\mu
\ge \frac{M_0}{M}|t|^\mu,
$$
for $|t|>M$.
 Consequently, from (G2), we have
$$
\frac{F(x,G^{-1}(t))}{|G^{-1}(t)|^{2\alpha}}
\ge \frac{M_0}{M}|t|^{\mu-2\alpha},\quad |t|>M.
$$
Notice  $\mu>2\alpha$, we get
 $$
\lim_{|t|\to \infty}\frac{F(x,G^{-1}(t))}{|G^{-1}(t)|^{2\alpha}}=+\infty,
$$
uniformly for $x\in\mathbb{R}^N$. Further, it follows from (G5)  that
  $$
\lim_{|t|\to \infty}\frac{F(x,t)}{|t|^{2\alpha}}=+\infty,
$$
uniformly for $x\in\mathbb{R}^N$. Consequently, the assumption (A7) implies (A5).
 The proof is complete.
\end{proof}

In the next theorem, we use the assumption
\begin{itemize}
\item[(A8)] $F(x,1)>0$ for any $x\in \mathbb{R}^N$, and there exists a constant
$\sigma>2\alpha$ such that any $c>1$,
$$
F(x,ct)\ge c^\sigma F(x,t), \quad  (x,t)\in\mathbb{R}^N\times\mathbb{R}.
 $$
\end{itemize}

\begin{theorem} \label{thm3.2}
Suppose that {\rm (A1)--(A4), (A6), (A8)}  are satisfied. Then \eqref{4} admits
a sequence of weak solutions $\{u_n\}\subset E$ such that
$\|u_n\|\to \infty$ and $J(u_n)\to \infty$ as $n\to \infty$.
\end{theorem}

\begin{proof}
 We consider that (A8) implies (A5). In fact,
 for any $x\in\mathbb{R}^N$ and $|s|>1$, we have
$$
F(x,|s|)\ge |s|^\sigma F(x,1).
$$
Consequently,
$$
\frac{F(x,s)}{|s|^{2\alpha}}\ge |s|^{\sigma-2\alpha} F(x,1).
$$
It follows from  $\sigma>2\alpha$ and  $F(x,1)>0$ that
 $$
\lim_{|s|\to \infty}\frac{F(x,s)}{|s|^{2\alpha}}=+\infty,
$$
uniformly for $x\in\mathbb{R}^N$. Consequently, the assumption {(A8)} 
implies {(A5)}. The proof is complete.
\end{proof}

In the next theorem, we use the assumption
\begin{itemize}
\item[(A9)] Assume $\widetilde{F}(x,t)=\frac14f(x,t)t-F(x,t)\ge0$,
and there exist constants $c>0$ and $\frac{2^*}{2^*-1}<\sigma<2$ such that
$$
 \widetilde{F}(x,t)\ge c |\frac{F(x,t)}{ t}|^\sigma, \quad
  (x,t)\in\mathbb{R}^N\times\mathbb{R}\quad  \text{with $t$ large enough}.
$$
\end{itemize}

\begin{theorem} \label{thm3.3}
Suppose that {\rm (A1)--(A3), (A5), (A9)}  are satisfied.
Then   \eqref{4} admits  a sequence of weak solutions $\{u_n\}\subset E$ 
such that $\|u_n\|\to \infty$ and $J(u_n)\to \infty$ as $n\to \infty$.
\end{theorem}

Following the method in \cite{15,9}, the theorem above can be obtained directly.

\subsection*{Acknowledgments}
 The authors were supported financially by the National Natural
Science Foundation of China (11571296).

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