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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 66, pp. 1--12.\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-2016/66\hfil Multiple homoclinic solutions]
{Multiple homoclinic solutions for  superquadratic Hamiltonian systems}

\author[W. Jiang, Q. Zhang \hfil EJDE-2016/66\hfilneg]
{Wei Jiang, Qingye Zhang}

\address{Wei Jiang \newline
Department of Mathematics,
Jiangxi Normal University, Nanchang 330022, China}
\email{jiangweijw1991@163.com}

\address{Qingye Zhang (corresponding author)\newline
Department of Mathematics,
Jiangxi Normal University, Nanchang 330022, China}
\email{qingyezhang@gmail.com}

\thanks{Submitted December 14, 2015. Published March 10, 2016.}
\subjclass[2010]{34C37, 35A15, 37J45}
\keywords{Hamiltonian system; Homoclinic solution; variational method}

\begin{abstract}
 In this article we study the  existence of infinitely many homoclinic
 solutions for a class of second-order Hamiltonian systems
 $$
 \ddot{u}-L(t)u+W_u(t,u)=0, \quad \forall t\in\mathbb{R},
 $$
 where $L$ is not required to be either uniformly positive
 definite or coercive, and $W$ is superquadratic at infinity in $u$
 but does not need to satisfy the Ambrosetti-Rabinowitz superquadratic condition.

\end{abstract}

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


\section{Introduction and statement of main results}

We consider the second-order Hamiltonian system
\begin{equation}
\ddot{u}-L(t)u+W_u(t,u)=0,\quad\forall  t\in\mathbb{R},\label{HS}
\end{equation}
where $u=(u_1,\ldots,u_N)\in \mathbb{R}^N$,
$W\in C^1(\mathbb{R}\times \mathbb{R}^N , \mathbb{R})$,
$L\in C(\mathbb{R},\mathbb{R}^{N^2})$ is a symmetric
matrix-valued function, and $W_u(t,u)$ denotes the gradient of $W(t,u)$
with respect to $u$. Here, as usual, we say that a solution $u$
of \eqref{HS} is homoclinic (to 0) if
$u\in C^2(\mathbb{R},\mathbb{R}^N)$,
$u(t)\not\equiv 0$, $u(t)\to 0$ and
$\dot u(t)\to 0$ as $|t| \to \infty$.

As a special case of dynamical systems, Hamiltonian systems are very important
in the study of gas dynamics, fluid mechanics, relativistic mechanics and
nuclear physics. They also appear in the fields of biology and chemistry
(see, e.g.,\cite{GR}). It is well known that homoclinic solutions play an
important role in analyzing the chaos of
Hamiltonian systems. If a system has the transversely intersected homoclinic
solutions, then it must be chaotic. If it has the smoothly connected homoclinic
solutions, then it cannot stand the perturbation, and its perturbed system
probably produces chaotic phenomena. Therefore, it is of practical importance
and mathematical significance to consider the existence of homoclinic solutions
of Hamiltonian systems emanating from 0.

During the previous decades, the existence and multiplicity of
homoclinic solutions for \eqref{HS} have been extensively
investigated via variational methods ; see
\cite{AB,AC,A,CM,CT,C,CR,D,DG,F,FS,KL,KLL,LT,OW,OT,P,R,RT,SCN,WT,WZX,WL,
YH,YZ,ZL,ZT,ZY,ZXY,Z}
and the references therein.  These methods
have also been used in many related and similar problems
(see, e.g.,\cite{CRV,PR,RS,ZHY}). From the beginning, most of
them  treated the case
where $L$ and $W$ are either independent of $t$ or periodic
in $t$ (see \cite{AB,AC,A,CR,DG,F,FS,P,R,RT,WT}).
In this kind of problem, the function $L$ plays an
important role. If $L$ is neither a constant nor periodic, the
problem is quite different from the ones just described, because of
the lack of compactness of the Sobolev embedding.
After the work of
Rabinowitz and Tanaka \cite{RT}, many results  were obtained for the case 
where $L$ is neither a constant nor periodic.
(see, \cite{CM,CT,C,D,F,KL,KLL,LT,OW,OT,SCN,WT,WZX,WL,YH,YZ,ZL,ZT,ZY,ZXY,Z}).
However, except for \cite{WT}, in all these mentioned papers $L$ was always required
to satisfy either the uniform positive-definiteness condition:
\begin{itemize}
\item[(A1)] there exists $c_0>0$ such that
\[
L(t)u\cdot u\ge c_0|u|^2,\quad \forall  (t,u)\in \mathbb{R}\times\mathbb{R}^N
\]
\end{itemize}
or the coercivity condition:
\begin{itemize}
\item[(A2)] the smallest eigenvalue of $L$ tends to
$\infty$ as $|t|\to \infty$, i.e.,
\[
l(t)\equiv \inf_{u\in \mathbb{R}^N,\,|u|=1}L(t)u\cdot u \to \infty\quad
\text{as }|t|\to \infty,
\]
\end{itemize}
where $\cdot$ and $|\cdot|$ denote the standard inner product and 
the associated norm in $\mathbb{R}^N$ respectively.
Most of these known results were obtained for the case where $W$ 
is superquadratic at infinity in $u$ and satisfy the usual assumption:
\begin{itemize}
\item[(A3)] $\lim_{|u|\to 0}W(t,u)/|u|^2=0$ uniformly for $t\in \mathbb{R}$.
\end{itemize}
In this case, the well-known Ambrosetti-Rabinowitz superquadratic
condition was usually assumed on $W$ (see, e.g., \cite{CM,CT,D,KL,KLL,OW}).

In this article, we study the existence of infinitely many homoclinic
solutions for \eqref{HS} in the case where $L$ is unnecessarily required to be
either uniformly positive definite or coercive, and $W$ satisfies some weak
superquadratic condition at infinity with respect to $u$. Before presenting
our assumptions, we introduce some notation.
\smallskip
\noindent\textbf{Notation.}
For two $N\times N$ symmetric matrices $M_1$ and $M_2$, we say that $M_1\ge M_2$ if
\[
\min_{u\in \mathbb{R}^N,|u|=1}(M_1-M_2)u\cdot u\ge 0
\]
and that $M_1\not\ge M_2$ if $M_1\ge M_2$ does not hold.

We use the following assumptions:
\begin{itemize}
\item[(A4)] The smallest eigenvalue of $L(t)$ is bounded from below.

\item[(A5)] There exists a constant $r_0>0$ such that
\[
\lim_{|s|\to \infty} \operatorname{meas}(\{t\in (s-r_0,s+r_0): L(t)\not\ge MI_N\})=0,
\quad \forall M>0,
\]
where $meas$ denotes the Lebesgue measure in $\mathbb{R}$ and $I_N$ is
the identity matrix in $\mathbb{R}^N$.

\item[(A6)] $W(t,0)\equiv 0$, and there exist constants $c_1>0$ and $\nu>2$
such that
\[
|W_u(t,u)|\le c_1(|u|+|u|^{\nu-1}),\quad \forall  (t,u)\in
\mathbb{R}\times\mathbb{R}^N;
\]
\item[(A7)] $\lim_{|u|\to \infty}W(t,u)/|u|^2=\infty$  uniformly for
$t\in \mathbb{R}$.

\item[(A8)] There exists a constant $\vartheta\ge 1$ such that
\[
\vartheta\widetilde{W}(t,u)\ge \widetilde{W}(t,su),\quad \forall
(t,u)\in\mathbb{R}\times\mathbb{R}^N
\text{ and }s\in [0,1],
\]
where $\widetilde{W}(t,u):=W_u(t,u)\cdot u-2W(t,u)$.

\item[(A9)] $W(t,-u)=W(t,u)$ for all $(t,u)\in \mathbb{R}\times\mathbb{R}^N$.
\end{itemize}

Our main result reads as follows.

\begin{theorem}\label{SOHS}
Suppose that {\rm (A4)}, {\rm (A5)} and {\rm (A6)--(A9)} are satisfied.
Then \eqref{HS} possesses a sequence of homoclinic solutions  \{$u_k$\} satisfying
\[
\frac{1}{2}\int_\mathbb{R}(|\dot{u}_k|^2+L(t)u_k\cdot u_k)dt
-\int_\mathbb{R}W(t, u_k)dt\to \infty\quad\text{\rm as } k\to \infty.
\]
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
 It is easy to see that conditions (A4) and (A5) are weaker than the coercivity
 condition (A2). In our Theorem \ref{SOHS}, $L$ is unnecessarily
uniformly positive definite. Besides, the usual condition (A3) and the
well-known Ambrosetti-Rabinowitz superquadratic condition are not required
in our Theorem \ref{SOHS}. There are functions $L$ and $W$ which satisfy all
the conditions in our Theorem \ref{SOHS} but do not satisfy the corresponding
conditions in the aforementioned references for the superquadratic case.
For example, let
\begin{gather*}
L(t)=(|t|\sin ^2t-1)I_N, \\
W(t,u)=a(t)\big[|u|^2\ln (e+|u|)-\frac{1}{2}|u|^2+e|u|-e^2(\ln (e+|u|)-1)\big],
\end{gather*}
where $a$ is a continuous bounded function with positive lower bound,
then simple computation shows that they satisfy {\rm (A4), (A5)} and
{\rm(A6)--(A9)}. However, $L$ does not satisfy neither the uniform
 positive-definiteness  condition  (A1) nor the coercivity condition (A2).
Meanwhile neither the usual assumption (A3) nor the
Ambrosetti-Rabinowitz superquadratic assumption holds for $W$.
\end{remark}

\section{Variational setting and proof of the main result}

To prove our main result via the critical point theory, we need to establish
the variational setting for \eqref{HS}. Before this, we have the following result.

\begin{remark} \label{LgeIN} \rm
 From (A4) and (A6), we know that there exists a positive constant $l_0$
such that $L(t)+2l_0I_N\ge I_N$ for all $t \in\mathbb{R}$ and
$W(t,u)+l_0|u|^2\ge 0$ for all $(t,u)\in\mathbb{R}\times\mathbb{R}^N$.
Let $\overline{L}(t)=L(t)+2l_0I_N$ and $\overline{W}(t,u)=W(t,u)+l_0|u|^2$.
Consider the following Hamiltonian system
\begin{equation}\label{eqHS}
\ddot{u}-\overline{L}(t)u+\overline{W}_u(t,u)=0,\quad\forall  t\in\mathbb{R},
\end{equation}
then \eqref{eqHS} is equivalent to \eqref{HS}. Moreover, it is easy to check
that the hypotheses {\rm (A4)}, {\rm (A5)} and {\rm (A6)--(A9)} still
hold for $\overline{L}$ and $\overline{W}$ provided that those hold for
 $L$ and $W$. Hence, in what follows, we always assume without loss of generality
that $L(t)\ge I_N$ for all $t \in\mathbb{R}$ and $W(t,u)\ge 0$ for all
$(t,u)\in\mathbb{R}\times\mathbb{R}^N$.
\end{remark}

In view of Remark \ref{LgeIN},  we consider the space
$E:=\{u\in H^1(\mathbb{R},\mathbb{R}^N)|\int_\mathbb{R}L(t)u\cdot udt<\infty\}$
equipped with the following inner product
\[
(u,v)=\int_\mathbb{R}(\dot{u}\cdot\dot{v}+L(t)u\cdot v)dt.
\]
Then $E$ is a Hilbert space and we denote by $\|\cdot\|$ the associated norm.
Moreover, we write $E^*$ for the topological dual of $E$, and
$\langle\cdot, \cdot\rangle : E^*\times E\to \mathbb{R}$ for the dual pairing.
Evidently, $E$ is continuously embedded into $H^1(\mathbb{R},\mathbb{R}^N)$
and hence continuously embedded into $L^p\equiv L^p(\mathbb{R},\mathbb{R}^N)$
for $2\le p\le \infty$, i.e., there exists $\tau_p>0$ such
that
\begin{equation}\label{LpleE}
\|u\|_p\leq \tau_p\|u\|,\quad\forall  u\in E,
\end{equation}
where $\|\cdot\|_p$ denotes the usual norm in $L^p$ for all $2\le p\le \infty$.
In fact, we further have the following lemma.

\begin{lemma}\label{EcpinLp}
If $L$ satisfies {\rm(A4)} and {\rm(A5)}, then $E$ is compactly embedded into
$L^p$ for $2\le p< \infty$.
\end{lemma}

\begin{proof}
 Let $\{u_n\}\subset E$ be a bounded sequence such that $u_n\rightharpoonup u$ in $E$.
 We will show that $u_n\to u$ in $L^p$ for $2\le p< \infty$. By the interpolation
inequality we only need to consider the case $p=2$. Suppose, without loss of
generality, that $u_n\rightharpoonup 0$ in $E$. The Sobolev embedding theorem
implies $u_n\to 0$ in $L^2_{loc}(\mathbb{R},\mathbb{R}^N)$. Thus it suffices to
show that, for any $\epsilon>0$, there is $r>0$ such that
$\int_{\mathbb{R}\backslash(-r,r)}|u_n|^2dt<\epsilon$. For any
$s\in \mathbb{R}$, we denote by $\mathcal{B}_{r_0}(s)$ the interval in
$\mathbb{R}$ centered at $s$ with radius $r_0$, i.e.,
$\mathcal{B}_{r_0}(s):=(s-r_0,s+r_0)$, where $r_0$ is the constant given in (A5).
Let $\{s_i\}\subset \mathbb{R}$ be a sequence of points such that
 $\mathbb{R}=\cup_{i=1}^\infty \mathcal{B}_{r_0}(s_i)$ and each
$t\in \mathbb{R}$ is contained in at most two such intervals. For any
$r>0$ and $M>0$, let
\begin{gather*}
\mathcal {C}(r,M)=\{t\in \mathbb{R}\setminus(-r,r):L(t)\ge MI_N\},\\
\mathcal {D}(r,M)=\{t\in \mathbb{R}\setminus(-r,r):L(t)\not\ge MI_N\}.
\end{gather*}
Then
\[
\int_{\mathcal {C}(r,M)}|u_n|^2dt
\le \frac{1}{M}\int_{\mathcal {C}(r,M)}L(t)u_n\cdot u_ndt
\le \frac{1}{M}\int_\mathbb{R}L(t)u_n\cdot u_ndt,
\]
and this can be made arbitrarily small by choosing $M$ large. Also for a fixed $M>0$,
\begin{align*}
\int_{\mathcal {D}(r,M)}|u_n|^2dt
&\le \sum_{i=1}^\infty\int_{\mathcal {D}(r,M)\cap \mathcal{B}_{r_0}(s_i)}|u_n|^2dt
\\
&\le \sum_{i=1}^\infty
 \Big(\int_{\mathcal {D}(r,M)\cap \mathcal{B}_{r_0}(s_i)}
|u_n|^4dt\Big)^{1/2}(\operatorname{meas}(\mathcal {D}(r,M)
 \cap \mathcal{B}_{r_0}(s_i)))^{1/2}\\
&\le \varepsilon_r\sum_{i=1}^\infty
\Big(\int_{\mathcal{B}_{r_0}(s_i)}|u_n|^4dt\Big)^{1/2}\\
&\le c\varepsilon_r\sum_{i=1}^\infty\int_{\mathcal{B}_{r_0}(s_i)}
(|\nabla u_n|^2+|u_n|^2)dt\\
&\le 2c\varepsilon_r\int_\mathbb{R}(|\nabla u_n|^2+|u_n|^2)dt
\end{align*}
for some constant $c>0$, where
$\varepsilon_r=\sup_{i\in \mathbb{N}} (\operatorname{meas}(\mathcal {D}(r,M)
\cap \mathcal{B}_{r_0}(s_i)))^{1/2}$. By (A5), $\varepsilon_r\to 0$ as $r\to\infty$.
Noting that $\{u_n\}$ is bounded in $E$, we can make this term small by choosing $r$
large. This completes the proof.
\end{proof}

For later use, we give the following two technical lemmas.

\begin{lemma}\label{forcpt}
Let {\rm(A4)}, {\rm(A5)} and  {\rm (A6)} be satisfied.
If $u_n\rightharpoonup u$ in $E$, then
\[
\int_{\mathbb{R}}|W_u(t, u_n)-W_u(t, u)|^2dt\to 0\quad\text{ as } n\to \infty.
\]
\end{lemma}

\begin{proof}
Arguing indirectly, we assume by Lemma \ref{EcpinLp} that there exists a
subsequence $\{u_{n_k}\}_{k\in \mathbb{N}}$ such that
\begin{equation}\label{unkL2ae}
u_{n_k}\to u\text{ in } L^2 \text{ and } L^{2\nu-2},\quad
\text{and }u_{n_k}\to u \text{ a.e. in $\mathbb{R}$ as } k\to \infty\,;
\end{equation}
and
\begin{equation}\label{Wuunku}
\int_{\mathbb{R}}|W_u(t, u_{n_k})-W_u(t, u)|^2dt\ge \epsilon_0,\quad
\forall k\in \mathbb{N}
\end{equation}
for some $\epsilon_0>0$. Passing to a subsequence if necessary, we may assume
by \eqref{unkL2ae} that $\sum_{k=1}^\infty\|u_{n_k}-u\|_2<\infty$
and $\sum_{k=1}^\infty\|u_{n_k}-u\|_{2\nu-2}<\infty$.
 Let $w(t)=\sum_{k=1}^\infty|u_{n_k}(t)-u(t)|$ for all $t\in \mathbb{R}$,
then $w\in L^2\cap L^{2\nu-2}$. By (A6), for all $k\in \mathbb{N}$ and
$t\in \mathbb{R}$, we have
\begin{align*}
|W_u(t,u_{n_k})-W_u(t,u)|^2
&\le c_1^2(|u_{n_k}|+|u|+|u_{n_k}|^{\nu-1}+|u|^{\nu-1})^2\\
&\le 4c_1^2(|u_{n_k}|^2+|u|^2+|u_{n_k}|^{2\nu-2}+|u|^{2\nu-2})\\
&\le 12c_1^2(|u_{n_k}-u|^2+|u|^2)+4^\nu c_1^2(|u_{n_k}-u|^{2\nu-2}+|u|^{2\nu-2})\\
&\le 12c_1^2(|w|^2+|u|^2)+4^\nu c_1^2(|w|^{2\nu-2}+|u|^{2\nu-2}).
\end{align*}
Combining this and \eqref{unkL2ae}, by Lebesgue's Dominated Convergence Theorem,
we have
\[
\lim_{k\to \infty}\int_{\mathbb{R}}|W_u(t,u_{n_k})-W_u(t,u)|^2dt=0,
\]
which contradicts \eqref{Wuunku}. The proof is complete.
\end{proof}

\begin{lemma}\label{measest}
For any finite dimensional subspace $F\subset E$ there exists a constant $\epsilon
>0$ such that
\[
\operatorname{meas}(\{t\in \mathbb{R}:|u(t)|\ge \epsilon \|u\|\})\ge
\epsilon,\quad\forall  u\in F\setminus \{0\}.
\]
\end{lemma}

\begin{proof}
We argue indirectly. Assume for any $n\in \mathbb{N}$,
there exists $u_n\in F\setminus \{0\}$ such that
\[
\operatorname{meas}(\{t\in \mathbb{R}:|u_n(t)|\ge \|u_n\|/n\})< 1/n.
\]
Let $v_n=u_n/\|u_n\|\in F$ for each $n\in\mathbb{N}$, then we have
$\|v_n\|=1$ and
\begin{equation}\label{mv1n}
\operatorname{meas}(\{t\in \mathbb{R}:|v_n(t)|\ge 1/n\})< 1/n.
\end{equation}
Passing to a subsequence if necessary, we may assume $v_n\to v_0$ in
$E$ for some $v_0\in F$ since $F$ is of finite dimension.
Combining this and \eqref{LpleE}, we have
\begin{equation}\label{vnv0L2}
\int_\mathbb{R}|v_n-v_0|^2dt\to 0\quad \text{as } n\to \infty.
\end{equation}
Noting that $\|v_0\|=1$, there must exists a constant $\delta_0>0$ such that
\begin{equation}\label{mvo}
\operatorname{meas}(\{t\in \mathbb{R}:|v_0(t)|\ge \delta_0 \})\ge \delta_0.
\end{equation}
Otherwise, for each fixed $n\in \mathbb{N}$, we have
\[
\operatorname{meas}\big(\{t\in \mathbb{R}:|v_0(t)|
\ge \frac{1}{n} \}\big)
\le \operatorname{meas}\big(\{t\in \mathbb{R}:|v_0(t)|
\ge \frac{1}{m}\}\big)\le \frac{1}{m},\quad \forall m\ge n.
\]
Lettin $m\to \infty$, we obtain
$\operatorname{meas}(\{t\in \mathbb{R}:|v_0(t)|\ge \frac{1}{n} \})=0$.
Consequently,
\begin{align*}
0&\le \operatorname{meas}(\{t\in \mathbb{R}:|v_0(t)|\neq 0 \})\\
&= \operatorname{meas}\big(\cup_{n=1}^\infty\{t\in \mathbb{R}:|v_0(t)|
 \ge \frac{1}{n} \}\big)\\
&\le \sum_{n=1}^\infty \operatorname{meas}
\big(\{t\in \mathbb{R}:|v_0(t)|\ge \frac{1}{n} \}\big)=0,
\end{align*} 
which yields $v_0= 0$, a contradiction to $\|v_0\|=1$.
Thus \eqref{mvo} holds. Set
$\mathcal{I}_0=\{t\in \mathbb{R}:|v_0(t)|\ge \delta_0 \}$, where $\delta_0$
is the constant given in \eqref{mvo}.
Also, for any $n\in \mathbb{N}$, let
\[
\mathcal {I}_n=\{t\in \mathbb{R}:|v_n(t)|< 1/n\}\quad \text{and}\quad
\mathcal{I}_n^c=\mathbb{R}^N\setminus \mathcal{I}_n=\{t\in \mathbb{R}:|v_n(t)|\ge
1/n\}.
\]
Then for $n$ large enough, by \eqref{mv1n} and \eqref{mvo}, we have
\[
\operatorname{meas}(\mathcal{I}_n\cap\mathcal{I}_0) \ge \operatorname{meas}(\mathcal{I}_0)-\operatorname{meas}(\mathcal{I}_n^c)
\ge \delta_0-1/n\ge \delta_0/2.
\]
Consequently, for  $n$ large enough, there holds
\begin{align*}
\int_\mathbb{R}|v_n-v_0|^2dt&\ge
\int_{\mathcal{I}_n\cap\mathcal{I}_0}|v_n-v_0|^2dt\\
&\ge \int_{\mathcal{I}_n\cap\mathcal{I}_0}(|v_0|-|v_n|)^2dt \\
&\ge (\delta_0-1/n)^2 \operatorname{meas}(\mathcal{I}_n\cap\mathcal{I}_0)\\
&\ge \delta_0^3/8>0.
\end{align*}
This is in contradiction to \eqref{vnv0L2}. The proof is complete.
\end{proof}

Now we can define the variational functional $\Phi$ associated with \eqref{HS} by
\begin{equation}\label{Phidef}
\begin{aligned}
\Phi (u)&=\frac{1}{2}\int_\mathbb{R}(|\dot{u}|^2+L(t)u\cdot u)dt
-\int_\mathbb{R}W(t, u)dt \\
&=\frac{1}{2}\|u\|^2-\int_\mathbb{R}W(t, u)dt.
\end{aligned}
\end{equation}
By (A6), we have
\begin{equation}\label{West}
|W(t,u)|\le c_1(|u|^2+|u|^\nu) ,\quad\forall (t,u)\in \mathbb{R}\times \mathbb{R}^{N}.
\end{equation}
This and \eqref{LpleE} imply that $\Phi $ is well
defined on $E$. Furthermore, a standard argument (see, e.g., \cite{RT})
shows that $\Phi \in C^1(E,\mathbb{R})$ with the Frech\'{e}t derivative given by
\begin{equation}\label{Phider}
\langle\Phi '(u),v\rangle=(u,v)-\int_\mathbb{R}W_u(t, u)\cdot vdt,\quad
\forall  u, v\in E,
\end{equation}
and nontrivial critical points of $\Phi $ are homoclinic solutions of \eqref{HS}.

To study the critical points of the variational functional $\Phi$ associated
 with \eqref{HS}, we need the following variant fountain theorem established in
\cite{Z2}.

Let $E$ be a Banach space with the norm $\|\cdot\|$ and
$E=\overline{\oplus_{j\in \mathbb{N}}X_j}$ with $\dim X_j<\infty$
for any $j\in \mathbb{N}$. Set $Y_k=\oplus_{j=1}^k X_j$ and
$Z_k=\overline{\oplus_{j=k}^\infty X_j}$. Consider the following
$C^1$-functional $\Phi _\lambda: E\to \mathbb{R}$
defined by
\[
\Phi _\lambda(u):=A(u)-\lambda B(u),\quad  \lambda\in [1,2].
\]
\begin{theorem}[{\cite[Theorem 2.1]{Z2}}]\label{ssvfthm}
Assume that the above functional $\Phi _\lambda$ satisfies
\begin{itemize}

\item[(A10)] $\Phi _\lambda$ maps bounded sets to
bounded sets for $\lambda\in [1,2]$, and
$\Phi _\lambda(-u)=\Phi _\lambda(u)$ for all $(\lambda,
u)\in [1,2]\times E$,

\item[(A11)] $B(u)\ge 0$ for all $u\in E${\rm ,}
and $A(u)\to \infty$ or $B(u)\to \infty$ as $\|u\|\to \infty$,

\item[(A12)] There exist $\rho_k>\sigma_k>0$ such that
\[
\alpha_k(\lambda):=\inf_{u\in
Z_k,\,\|u\|=\sigma_k}\Phi _\lambda(u)>\beta_k(\lambda):=\max_{u\in
Y_k,\,\|u\|=\rho_k}\Phi _\lambda(u),\quad \forall
\lambda\in[1,2].
\]
\end{itemize}
Then
\[
\alpha_k(\lambda)\le
\zeta_k(\lambda):=\inf_{\gamma\in \Gamma_k}\max_{u\in
B_k}\Phi _\lambda(\gamma(u)),\quad \forall  \lambda\in[1,2],
\]
where $B_k=\{u\in Y_k: \|u\|\le \rho_k\}$ and
$\Gamma_k:=\{\gamma\in C(B_k, E): \gamma\text{ is odd, }
\gamma|_{\partial B_k}=id\}$. Moreover, for almost every
$\lambda\in[1,2]$, there exists a sequence
$\{u_m^k(\lambda)\}_{m=1}^\infty$ such that
\[
\sup_m \|u_m^k(\lambda)\|<\infty,\quad \Phi '_\lambda(u_m^k(\lambda))\to
0,\quad \Phi _\lambda(u_m^k(\lambda))\to
\zeta_k(\lambda)\quad \text{as } m\to \infty.
\]
\end{theorem}

Choose an orthonormal basis $\{e_j: j\in\mathbb{N}\}$ of $E$ and let
$X_j={\rm span}\{e_j\}$ for all $j\in\mathbb{N}$. Define the functionals
 $A$, $B$ and $\Phi _{\lambda}$ on our
working space $E$ by
\begin{gather}\label{ABdef}
A(u)=\frac{1}{2}\|u\|^2,\quad
B(u)={\it\Psi}(u)=\int_\mathbb{R} W(t,u)dt, \\
\label{Philamdef}
\Phi _{\lambda}(u)=A(u)-\lambda B(u)
=\frac{1}{2}\|u\|^2-\lambda\int_\mathbb{R} W(t,u)dt
\end{gather}
for all $u\in E$ and
$\lambda\in [1,2]$. Note that $\Phi _1=\Phi $,
where $\Phi $ is the functional defined in \eqref{Phidef}. Then we know that
$\Phi _{\lambda}\in C^1(E, \mathbb{R})$ for all
$\lambda\in [1,2]$ and
\begin{equation}\label{Philamder}
\langle\Phi '_\lambda(u),v\rangle=(u,v)-\lambda\int_\mathbb{R}W_u(t, u)\cdot vdt,
\quad \forall  u, v\in E.
\end{equation}

Before applying Theorem \ref{ssvfthm} to prove our main result, we need to
establish the following two lemmas.

\begin{lemma}\label{forF31}
Assume {\rm (A4)}, {\rm (A5)} and {\rm (A6)} hold. Then there exists a
positive integer $k_1$ and a sequence $\sigma_k\to \infty$ as
$k\to \infty$ such that
\[
\alpha_k(\lambda):=\inf_{u\in Z_k,\,\|u\|=\sigma_k}\Phi _\lambda(u)> 0,\quad \forall  k\ge k_1,
\]
where
$Z_k=\overline{\oplus_{j=k}^\infty
X_j}=\overline{{\operatorname{span}\{e_{k},\ldots\}}}$ for all $k\in
\mathbb{N}$.
\end{lemma}

\begin{proof}
Note first that \eqref{West} and
\eqref{Philamdef} imply
\begin{equation} \label{Philamge}
\begin{aligned}
\Phi _{\lambda}(u)
&\ge\frac{1}{2}\|u\|^2-2\int_\mathbb{R} W(t,u)dt \\
&\ge\frac{1}{2}\|u\|^2-2c_1(\|u\|_2^2+\|u\|_\nu^\nu),\quad
\forall (\lambda,u)\in[1,2]\times E.
\end{aligned}
\end{equation}
For each $k\in\mathbb{N}$, let
\begin{equation}\label{l2lnu}
\ell_2(k)=\sup_{u\in Z_k,\,\|u\|=1}\|u\|_2\quad\text{and}
\quad \ell_\nu(k)=\sup_{u\in Z_k,\,\|u\|=1}\|u\|_\nu.
\end{equation}
Since $E$ is compactly embedded into both $L^2$ and $L^\nu$ by Lemma \ref{EcpinLp},
then there hold (cf. \cite{W})
\begin{equation}\label{l2lnuto0}
\ell_2(k)\to 0,\quad \ell_\nu(k)\to 0 \quad \text{as } k\to \infty.
\end{equation}
Combining \eqref{Philamge} and \eqref{l2lnu}, we have
\begin{equation}\label{gel2lnu}
\Phi _{\lambda}(u)\ge
\frac{1}{2}\|u\|^2-2c_1\ell_2^2(k)\|u\|^2-2c_1\ell_\nu^\nu(k)\|u\|^\nu,
\quad\forall  (\lambda,u)\in [1,2]\times Z_k.
\end{equation}
In view of \eqref{l2lnuto0}, there exists a positive integer $k_1$
such that
\begin{equation}\label{ell214}
2c_1\ell_2^2(k)\le 1/4,\quad\forall k\ge k_1.
\end{equation}
For each $k\ge k_1$, choose
\begin{equation}\label{sigmakdef}
\sigma_k:=(16c_1\ell_\nu^\nu(k))^{1/(2-\nu)}.
\end{equation}
Then it follows from \eqref{l2lnuto0} that
\begin{equation}\label{sigmakinfty}
\sigma_k\to +\infty \quad \text{as } k\to \infty
\end{equation}
since $\nu>2$.
Besides, by  \eqref{gel2lnu}--\eqref{sigmakdef}, direct computation
shows
\[
\alpha_k(\lambda):=\inf_{u\in
Z_k,\,\|u\|=\sigma_k}\Phi _\lambda(u)\ge \sigma_k^2/8
>0,\quad\forall k\ge k_1.
\]
The proof is complete.
\end{proof}

\begin{lemma}\label{forF32}
Suppose that {\rm (A4)}, {\rm (A5)}, {\rm (A6)} and {\rm (A7)} are satisfied. 
Then for the positive integer $k_1$ and the sequence $\{\sigma_k\}$ obtained 
in Lemma \ref{forF31}, there exists $\rho_k>\sigma_k$ for each $k\ge
k_1$ such that
\[
\beta_k(\lambda):=\max_{u\in
Y_k,\,\|u\|=\rho_k}\Phi _\lambda(u)<0,
\]
where $Y_k=\oplus_{j=1}^k X_j=\operatorname{span}\{e_1,\ldots, e_k\}$ 
for all $k\in \mathbb{N}$.
\end{lemma}

\begin{proof} 
Note that $Y_k$ is finite dimensional for each $k\in\mathbb{N}$. 
Then by Lemma \ref{measest}, for each $k\in \mathbb{N}$, there exists a 
constant $\epsilon_k>0$ such that
\begin{equation}\label{Iuk}
\operatorname{meas}(\mathcal{I}_u^k)\ge \epsilon_k, \quad
\forall u\in Y_k\setminus \{0\},
\end{equation}
where $\mathcal{I}_u^k:=\{t\in \mathbb{R}:|u(t)|\ge \epsilon_k\|u\|\}$ for
all $k\in \mathbb{N}$ and $u\in Y_k\setminus \{0\}$. By (A7), for
each $k\in \mathbb{N}$, there exists a constant $b_k>0$ such that
\begin{equation}\label{geu2}
W(t,u)\ge u^2/\epsilon_k^3, \quad\forall t\in\mathbb{R}\text{ and }|u|\ge b_k.
\end{equation}
Combining \eqref{Philamdef}, \eqref{Iuk} and \eqref{geu2}, for any 
$k\in \mathbb{N}$ and $\lambda\in [1,2]$, we have
\begin{align}
\Phi _{\lambda}(u)&\le\frac{1}{2}\|u\|^2-\int_\mathbb{R} W(t,u)dt \\
&\le\frac{1}{2}\|u\|^2-\int_{\mathcal{I}_u^k}(|u|^2/\epsilon_k^3)dt \\
&\le\frac{1}{2}\|u\|^2-\epsilon_k^2\|u\|^2 \operatorname{meas}
(\Lambda_u^k)/\epsilon_k^3 \\
&\le\frac{1}{2}\|u\|^2-\|u\|^2=-\frac{1}{2}\|u\|^2\label{ssPhi-u2}
\end{align}
for all $u\in Y_k$ with $\|u\|\ge b_k/\epsilon_k$. Here we use the fact 
that $W(t,u)\ge 0$ for all $(t,u)\in\mathbb{R}\times\mathbb{R}^N$.
For each $k\ge k_1$, if we choose
$\rho_k>\max\{\sigma_k,b_k/\epsilon_k\}$,
then \eqref{ssPhi-u2} implies
\[
\beta_k(\lambda):=\max_{u\in
Y_k,\,\|u\|=\rho_k}\Phi _\lambda(u)\le -\rho_k^2/2<0.
\]
The proof is complete.
\end{proof}

Now we are in a position to give the proof of our main result.

\begin{proof}[Proof of Theorem \ref{SOHS}]
 Firstly,  from
\eqref{LpleE}, \eqref{West} and \eqref{Philamdef} it follows that
$\Phi _\lambda$ maps bounded sets to bounded sets uniformly for
$\lambda\in [1,2]$. Evidently, (A9) implies that 
$\Phi _\lambda(-u)=\Phi _\lambda(u)$ for all $(\lambda,
u)\in [1,2]\times E$. Thus (A10) holds. Next, using again the 
fact that $W(t,u)\ge 0$ 
for all $(t,u)\in\mathbb{R}\times\mathbb{R}^N$, we know that  (A11) 
holds by the definition of functional $A$ in \eqref{ABdef}. 
Finally, Lemma \ref{forF31} and Lemma
\ref{forF32} show that  (A12) holds
for all $k\ge k_1$, where $k_1$ is given in Lemma
\ref{forF31}. Therefore, for each $k\ge k_1$, by Theorem
\ref{ssvfthm}, for almost every $\lambda\in [1,2]$, there exists a sequence
$\{u_m^k(\lambda)\}_{m=1}^\infty\subset E$ such that
\begin{equation}\label{aelam}
\sup_m \|u_m^k(\lambda)\|<\infty,\;\Phi '_\lambda(u_m^k(\lambda))\to 0
\quad\text{and}\quad
 \Phi _\lambda(u_m^k(\lambda))\to \zeta_k(\lambda)\quad\text{as } m\to \infty,
\end{equation}
where
\[
\zeta_k(\lambda):=\inf_{\gamma\in \Gamma_k}\max_{u\in
B_k}\Phi _\lambda(\gamma(u)),\quad \forall  \lambda\in[1,2]
\]
with $B_k=\{u\in Y_k: \|u\|\le \rho_k\}$ and $\Gamma_k:=\{\gamma\in
C(B_k, E): \gamma\text{ is odd, } \gamma|_{\partial B_k}=id\}$.
From the proof of Lemma \ref{forF31}, we infer that
\begin{equation}\label{zetakin}
\zeta_k(\lambda)\in
\left[\bar{\alpha}_k,\bar{\zeta}_k\right],\quad\forall k\ge
k_1\text{ and }\lambda\in [1,2],
\end{equation}
where $\bar{\zeta}_k:=\max_{u\in B_k}\Phi _1(u)$ and
$\bar{\alpha}_k:=\sigma_k^2/4\to \infty$ as $k\to \infty$ by
\eqref{sigmakinfty}. In view of \eqref{aelam}, for each $k\ge k_1$, 
we can choose a sequence $\lambda_n\to 1$ (depending on $k$) and get 
the corresponding sequences satisfying
\begin{equation}\label{aelamn}
\sup_m \|u_m^k(\lambda_n)\|<\infty\quad\text{and}\quad
\Phi '_{\lambda_n}(u_m^k(\lambda_n))\to 0\quad\text{as } m\to \infty.
\end{equation}

\noindent\textbf{Claim 1.}  For each $\lambda_n$ given above, the sequence
$\{u_m^k(\lambda_n)\}_{m=1}^\infty$
has a strong convergent subsequence.

For notational simplicity, we will set $u_m=u_m^k(\lambda_n)$ for
$m\in \mathbb{N}$ throughout the proof of Claim 1. 
By \eqref{aelamn}, without loss of generality, we may assume that
\begin{equation}\label{weaklim}
u_m\rightharpoonup u\quad \text{as }m\to \infty
\end{equation}
for some $u\in E$. Invoking \eqref{Philamder}, we have
\begin{equation} \label{umtou}
\begin{aligned}
\|u_m-u\|^2
&=\langle\Phi '_{\lambda_n}(u_m), u_m-u\rangle
 -\langle\Phi '_{\lambda_n}(u), u_m-u\rangle \\
&\quad +\lambda_n\int_{\mathbb{R}}(W_u(t,u_n(t))-W_u(t,u))\cdot(u_m-u)dt.
\end{aligned}
\end{equation}
By \eqref{aelamn}, we have
\begin{equation}\label{firstterm}
\langle\Phi '_{\lambda_n}(u_m), u_m-u\rangle\to 0\quad \text{as }m\to \infty.
\end{equation}
Moreover, \eqref{weaklim} yields
\begin{equation}\label{secondterm}
\langle\Phi '_{\lambda_n}(u), u_m-u\rangle\to 0\quad \text{as }m\to \infty.
\end{equation}
By  \eqref{LpleE}, Lemma \ref{forcpt} and the H\"{o}lder inequality, we have
\begin{equation}
\begin{aligned}
&\big|\int_{\mathbb{R}}(W_u(t,u_n(t))-W_u(t,u))\cdot(u_m-u)dt\big| \\
&\le \Big(\int_{\mathbb{R}}|W_u(t, u_n)-W_u(t, u)|^2dt\Big)^{1/2}\|u_m-u\|_2 \\
&\le c_2\Big(\int_{\mathbb{R}}|W_u(t, u_n)-W_u(t, u)|^2dt\Big)^{1/2
}\|u_m-u\|\to 0\quad \text{as }m\to \infty,
\end{aligned} \label{thirdterm}
\end{equation}
where $c_2$ is the constant given in \eqref{LpleE}. Here we use the fact that
 $\{u_m\}$ is bounded in $E$. Combining  \eqref{umtou} and
\eqref{firstterm}--\eqref{thirdterm}, we obtain $u_m\to u$ in $E$.
Thus Claim 1 holds.

By Claim 1, without loss of generality, we may assume that
\begin{equation}\label{unk}
\lim_{m\to \infty}u_m^k(\lambda_n)=u_n^k\in E,\quad
\forall n\in\mathbb{N} \text{ and } k\ge k_1.
\end{equation}
This, \eqref{aelam} and \eqref{zetakin} imply
\begin{equation}\label{Phider0}
\Phi '_{\lambda_n}(u_n^k)=0,\quad \Phi _{\lambda_n}(u_n^k)\in
[\bar{\alpha}_k,\bar{\zeta}_k],\quad \forall  n\in
\mathbb{N} \text{ and } k\ge k_1.
\end{equation}

\noindent\textbf{Claim 2.}  For each $k\ge k_1$, the sequence
$\{u_n^k\}_{n=1}^\infty$ in \eqref{unk} is bounded.

As in the proof of Claim 1, for notational simplicity, we set $u_n=u_n^k$ 
for all $n\in \mathbb{N}$. We use a indirect argument.
If Claim 2 is not true, without loss of generality, we may assume that
\begin{equation}\label{inftyweak}
\|u_n\|\to \infty\quad \text{and}\quad 
w_n:=\frac{u_n}{\|u_n\|}\rightharpoonup w\in E\quad\text{as }n\to \infty.
\end{equation}
By \eqref{inftyweak} and Lemma \ref{EcpinLp}, passing to a subsequence 
if necessary, we have
\begin{gather}\label{wntowinLp}
w_n\to w\quad\text{in } L^p\quad\text{for } 2\le p<\infty, \\
\label{wntowae}
w_n(t)\to w(t)\quad\text{a.e. } t\in \mathbb{R}.
\end{gather}
When $w\neq 0$ occurs, $\Theta:=\{t\in \mathbb{R}: w(t)\ne 0\}$ 
 has a positive Lebesgue measure. By \eqref{inftyweak}, it holds that
\begin{equation}\label{unTheta}
u_n(t)\to \infty,\quad \forall  t\in \Theta.
\end{equation}
Combining \eqref{Philamdef}, \eqref{wntowae}, \eqref{unTheta} and 
(A7), by Fatou's Lemma, we have
\begin{align*}
\frac{1}{2}-\frac{\Phi _{\lambda_n}(u_n)}{\|u_n\|^2}
&=\lambda_n\int_\mathbb{R}\frac{W(t,u_n)}{\|u_n\|^2}dt\\
&\ge \int_\Theta|w_n|^2\frac{W(t,u_n)}{|u_n|^2}dt\to 
 +\infty\quad \text{as } n\to \infty,
\end{align*}
a contradiction to \eqref{Phider0} and \eqref{inftyweak}.

When $w=0$ occurs, as in \cite{J}, we choose a sequence 
$\{s_n\}\subset [0, 1]$ such that
\begin{equation}\label{sndef}
\Phi _{\lambda_n}(s_nu_n)=\max_{s\in[0,1]}\Phi _{\lambda_n}(su_n).
\end{equation}
For $M>0$, let $\widetilde{w}_n:=\sqrt{4M}w_n=\frac{\sqrt{4M}}{\|u_n\|}u_n$, 
then \eqref{wntowinLp} yields
\begin{equation}\label{wntildeto0}
\widetilde{w}_n\to \sqrt{4M}w=0\quad\text{in } L^p\quad\text{for } 
2\le p<\infty.
\end{equation}
This \eqref{West} and \eqref{wntowinLp} imply
\begin{equation}\label{Fto0}
\Big|\int_\mathbb{R}W(t,\widetilde{w}_n)dt\Big|
\le c_1\int_\mathbb{R}(|\widetilde{w}_n|^2+|\widetilde{w}_n|^\nu)dt\to 0
\quad \text{as } n\to \infty.
\end{equation}
Note that $0<\frac{\sqrt{4M}}{\|u_n\|}<1$ holds by \eqref{inftyweak} 
for $n$ large enough.
Combining this with \eqref{Philamdef} and \eqref{sndef}, we obtain
\begin{align*}
\Phi _{\lambda_n}(s_nu_n)
&\ge \Phi _{\lambda_n}(\widetilde{w}_n)\\
&=\frac{1}{2}\|\widetilde{w}_n\|^2-\lambda_n\int_\mathbb{R}W(t,\widetilde{w}_n)dt\\
&=2M-\lambda_n\int_\mathbb{R}W(t,\widetilde{w}_n)dt
\ge M.
\end{align*}
for $n$ large enough. It follows that 
$\lim_{n\to \infty}\Phi _{\lambda_n}(s_nu_n)=+\infty$. 
Observing that $\Phi _{\lambda_n}(0)=0$ and 
$\Phi _{\lambda_n}(u_n)\in [\bar{\alpha}_k,\bar{\zeta}_k$ in \eqref{Phider0}, 
we know that $s_n\in (0,1)$ in \eqref{sndef} for $n$ large enough. Hence,
\begin{equation}\label{snun}
0=s_n\frac{{\rm d}}{{\rm d}s}\Big|_{s=s_n}\Phi _{\lambda_n}(su_n)
=\langle\Phi '_{\lambda_n}(s_nu_n),s_nu_n\rangle.
\end{equation}
Combining \eqref{Philamdef}, \eqref{Philamder}, \eqref{Phider0}, \eqref{snun} 
and (A8), we have
\begin{align*}
\Phi _{\lambda_n}(u_n)
&=\Phi _{\lambda_n}(u_n)-\frac{1}{2}\langle\Phi '_{\lambda_n}(u_n),u_n\rangle
\\
&=\frac{\lambda_n}{2}\int_\mathbb{R}\widetilde{W}(t,u_n)dt\\
&\ge\frac{\lambda_n}{2\vartheta}\int_\mathbb{R}\widetilde{W}(t,s_nu_n)dt\\
&=\frac{1}{\vartheta}\Phi _{\lambda_n}(s_nu_n)-\frac{1}{2\vartheta}\langle\Phi '_{\lambda_n}(s_nu_n),s_nu_n\rangle\\
&=\frac{1}{\vartheta}\Phi _{\lambda_n}(s_nu_n)\to +\infty\quad \text{as } n\to \infty.
\end{align*}
where $\vartheta$ is the constant in (A8).
This also provides a contradiction to \eqref{Phider0}.
Thus Claim 2 is true.

In view of Claim 2 and \eqref{Phider0}, for each $k\ge k_1$,
using the similar arguments in the proof of Claim 1, we can also show that
the sequence $\{u_n^k\}_{n=1}^\infty$ has a strong convergent subsequence 
with the limit $u^k$ being
just a critical point of $\Phi =\Phi _1$. Evidently,
$\Phi (u^k)\in \left[\bar{\alpha}_k,\bar{\zeta}_k\right]$ for all $k\ge k_1$. Since
$\bar{\alpha}_k\to +\infty$ as $k\to \infty$ in \eqref{zetakin}, we
obtain infinitely many nontrivial critical points of $\Phi $.
Therefore, \eqref{HS} possesses infinitely many nontrivial
solutions. The proof of Theorem \ref{SOHS} is complete. 
\end{proof}

\subsection*{Acknowledgments}
 This work is partially supported by the NFSC (11201196, 11361029).


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