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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 128, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/128\hfil Homoclinic solutions]
{Homoclinic solutions for a class of  second order non-autonomous
systems}

\author[R. Yuan, Z. Zhang\hfil EJDE-2009/128\hfilneg]
{Rong Yuan, Ziheng Zhang}  % in alphabetical order

\address{Rong Yuan \newline
School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education,
Beijing 100875, China}
\email{ryuan@bnu.edu.cn}

\address{Ziheng Zhang \newline
School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education,
Beijing 100875, China}
\email{zhzh@mail.bnu.edu.cn}

\thanks{Submitted October 14, 2008. Published October 7, 2009.}
\thanks{Supported by National Natural Science Foundation of China and RFDP}
\subjclass[2000]{34C37, 35A15, 37J45}
\keywords{Homoclinic solutions; critical point; variational methods;
\hfill\break\indent mountain pass theorem}

\begin{abstract}
 This article concerns the existence of homoclinic solutions
 for the second order non-autonomous system
 $$
 \ddot q+A \dot q-L(t)q+W_{q}(t,q)=0,
 $$
 where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric
 positive definite matrix depending continuously on $t\in \mathbb{R}$,
 $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$.
 We assume that $W(t,q)$ satisfies the global Ambrosetti-Rabinowitz
 condition, that the norm of $A$ is sufficiently small and
 that $L$ and $W$ satisfy additional hypotheses. We prove the
 existence of at least one nontrivial homoclinic solution, and
 the existence of infinitely  many homoclinic solutions if $W(t,q)$
 is even in $q$. Recent results in the literature are generalized
 and improved.
\end{abstract}

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

\section{Introduction}

The purpose of this work is to study the existence of
{\it homoclinic } solutions for the second order non-autonomous
 system
\begin{equation}
\ddot q+A \dot q-L(t)q+W_{q}(t,q)=0,\label{DS}
\end{equation}
where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric
and positive definite matrix depending continuously on $t\in \mathbb{R}$,
$W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$. A solution
$q(t)$ of \eqref{DS} is called a homoclinic solution (to 0) if
$q\in C^2(\mathbb{R},\mathbb{R}^n)$, $q(t)\to 0$ and
$\dot q(t)\to 0$ as $t\to\pm\infty$. If $q(t)\not\equiv 0$, $q(t)$
is called a nontrivial homoclinic solution.


When $A=0$, \eqref{DS} is the second order Hamiltonian system.
Assuming that $L(t)$ and $W(t,q)$ are independent of $t$ or
$T$-periodic in $t$,  the existence of homoclinic
solutions for the Hamiltonian system \eqref{DS} has been studied
via critical point theory and variational methods,
see for instance \cite{Am,Caldiroli,Co2,Di2,Fl,Paturel,Rab3}
and the references therein; a more general case is considered in
\cite{Iz}. In this case, the existence of homoclinic solutions
can be obtained by taking the limit of periodic solutions of
approximating problems. If $L(t)$ and $W(t,q)$ are neither
independent of $t$ not  periodic in $t$, compactness arguments
derived from  Sobolev imbedding theorem are not
available for the study of \eqref{DS}, see
 \cite{Al,Ca,Di1,Korman,Lv,Om,Ou,Rab1} and the references therein.

When $A\neq 0$,  as far as we know,  the existence of homoclinic
solutions of \eqref{DS} has not been studied.
Our basic hypotheses on $L$ and $W$ are:
\begin{itemize}
\item [(H1)]  $L\in C(\mathbb{R},\mathbb{R}^{n^2})$, $L(t)$ is a symmetric
  and positive definite  matrix for all $t\in \mathbb{R}$, and there is a
  continuous function $\alpha:\mathbb{R}\to \mathbb{R}$
  such that $\alpha(t)>0$ for all $t\in \mathbb{R}$,
  $\big(L(t)q,q\big)\geq \alpha(t)|q|^2$,
  and $\alpha(t)\to +\infty$ as $|t|\to +\infty$.

\item[(H2)]There exists a constant $\mu>2$ such that for every
 $t\in \mathbb{R}$ and $q\in \mathbb{R}^n\backslash\{0\}$
  $$
0<\mu W(t,q)\leq\big(W_{q}(t,q),q\big).
  $$

\item[(H3)] $W_q(t,q)=o(|q|)$ as $|q|\to 0$ uniformly with respect
   to $t\in \mathbb{R}$.

\item[(H4)]There exists $\overline{W}\in C(\mathbb{R}^n,\mathbb{R})$
such that
  $|W_q(t,q)|\leq | \overline{W}(q)|$
for every $t\in \mathbb{R}$ and $q\in \mathbb{R}^n$.
\end{itemize}

\begin{remark}\label{rem 1.1}
{\rm
From (H1), we see that there is a constant $\beta>0$ such that
\begin{equation}\label{1.1}
\big(L(t)q,q\big)\geq \beta|q|^2 \quad \text{for all $t\in \mathbb{R}$
and $q\in \mathbb{R}^n$}.
\end{equation}
(H2) is called the global Ambrosetti-Rabinowitz condition due to
 Ambrosetti and Rabinowitz (e.g., \cite{AmR}).
Combining (H2) with (H3), we see that
$W(t,q)\geq 0$ for all $(t,q)\in \mathbb{R}\times \mathbb{R}^{n}$,
$W(t,0)=0$, $W_q(t,0)=0$. Moreover, $W(t,q)=o(|q|^2)$ as $|q|\to 0$
uniformly with respect to $t$, which implies that for any
$\varepsilon>0$ there is $\delta>0$ such that
\begin{equation}\label{1.2}
W(t,q)\leq \varepsilon |q|^2\quad \text{for }
 (t,q)\in \mathbb{R}\times \mathbb{R}^n,\; |q|\leq \delta.
\end{equation}
}\end{remark}

In addition, we need the following hypothesis on  $A$.
\begin{itemize}
\item[(H5)] $\|A\|<\sqrt{\beta}$, where $\beta$ is defined
in  \eqref{1.1}.

\end{itemize}
 Now  we   state our main result.
\begin{theorem}\label{Thm 1.1}
Assume {\rm (H1)--(H5)}. Then \eqref{DS} possesses at least one
nontrivial homoclinic solution. Moreover, if we assume that
$W(t,q)$ is even in $q$; i.e.,
\begin{itemize}
\item[(H6)] $W(t,-q)=W(t,q)$  for all $t\in \mathbb{R}$ and
  $q\in \mathbb{R}^n$,
\end{itemize}
then \eqref{DS} has infinitely many distinct homoclinic solutions.
\end{theorem}

\begin{remark}
{\rm  From Remark \ref{rem 1.1}, we know that there exists $\beta>0$
such that \eqref{1.1} holds. However, since we do not have an
explicit estimate on $\beta$, we simply assume that  $\|A\|$
is sufficiently small.
Furthermore, when $A=0$, our main result  is just
\cite[Theorems 1 and 2]{Om}.
}\end{remark}

To overcome the lack of compactness in standard Sobolev imbedding
theorems,  we employ a compact imbedding theorem obtained in \cite{Om}.
In Section 2 we state and prove preliminary results. Section 3
is devoted to the proof of  Theorem \ref{Thm 1.1}.


\section{Preliminaries}

Let
$$
E=\big\{q\in
H^1(\mathbb{R},\mathbb{R}^n):\int_{\mathbb{R}}\big[|\dot q(t)|^2+\big(L(t)q(t),q(t)\big)
\big]dt<+\infty\big\}.
$$
This vector space is a Hilbert space when endowed with the inner
product
$$
(x,y)=\int_{\mathbb{R}}\big[\big(\dot x(t),\dot y(t)\big)+\big(L(t)x(t),y(t)\big)\big]dt
$$
and the corresponding norm $\|x\|^2=(x,x)$. Note that
$$
E\subset H^1(\mathbb{R},\mathbb{R}^n)\subset L^p(\mathbb{R},\mathbb{R}^n)
$$
for all $p\in[2,+\infty]$ with the imbedding being continuous.
In particular, for $p=+\infty$, there exists a constant $C>0$
such that
\begin{equation}\label{2.1}
\|q\|_{\infty}\leq C\|q\|, \quad \forall q\in E.
\end{equation}
Here $L^p(\mathbb{R},\mathbb{R}^n)$ ($2\leq p <+\infty$) and
$H^1(\mathbb{R},\mathbb{R}^n)$ denote the Banach spaces of functions on
$\mathbb{R}$ with values in $\mathbb{R}^n$ under the norms
$$
\|q\|_p:=\big(\int_{\mathbb{R}}|q(t)|^p dt\big)^{1/p}\quad \text{and} \quad
\|q\|_{H^1}:= \big(\|q\|_2^2+\|\dot q\|_2^2\big)^{1/2}
$$
respectively. $L^{\infty}(\mathbb{R},\mathbb{R}^n)$ is the Banach space
of essentially bounded functions from $\mathbb{R}$ into
$\mathbb{R}^n$ equipped with the norm
$$
\|q\|_{\infty}:=\text{ess} \sup\{|q(t)|: t\in \mathbb{R} \}.
$$

\begin{lemma}[{\cite[Lemma 1]{Om}}] \label{lem 2.1}
Assume  $L$ satisfies {\rm (H1)}. Then the embedding of $E$
in $L^2(\mathbb{R},\mathbb{R}^n)$  is compact.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2]{Om}}] \label{lem 2.2}
Assume {\rm  (H1), (H3), (H4)}. If $q_k\rightharpoonup q_0$ (weakly)
in $E$, then $W_q(t,q_k)\to W_q(t,q_0)$ in $L^2(\mathbb{R},\mathbb{R}^n)$.
\end{lemma}

\begin{lemma}\label{lem 2.3}
Under Assumption {\rm (H2)}, for every $t\in\mathbb{R}$, we have
\begin{gather}
W(t,q)\leq W\Big(t,\frac{q}{|q|}\Big)|q|^{\mu},\quad\text{if } 0<|q|\leq 1, \\
W(t,q)\geq W\Big(t,\frac{q}{|q|}\Big)|q|^{\mu},\quad\text{if } |q|\geq 1.
\end{gather}
\end{lemma}

\begin{proof}
It suffices to show that for every $q\neq 0$ and $t\in \mathbb{R}$
the function $(0,\infty)\ni \zeta\to W(t,\zeta^{-1}q)\zeta^{\mu}$
is non-increasing, which is an immediate consequence of (H2).
\end{proof}

\begin{remark}\label{rem 2.1}
{\rm From Lemma \ref{lem 2.3}, we see that there exists
$\alpha_0(t)>0$ such that
 $$
W(t,q)\geq \alpha_0(t)|q|^{\mu}\quad \text{for all }
 (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\; |q|\geq 1.
$$
}\end{remark}

Now we introduce more notation and some definitions.
Let $\mathcal{B}$ be a real Banach space,
$I\in C^1(\mathcal{B},\mathbb{R})$, which means that $I$ is a
continuously Fr\'echet-differentiable functional defined on
$\mathcal{B}$.

\begin{definition}[\cite{Rab2}]
{\rm $I\in C^1(\mathcal{B},\mathbb{R})$ is said to satisfy the (PS)
 condition if any sequence
$\{u_{j}\}_{j\in \mathbb{N}}\subset \mathcal{B}$, for which
$\{I(u_{j})\}_{j\in \mathbb{N}}$ is bounded and $I'(u_{j})\to 0$
as $j\to +\infty$, possesses a convergent subsequence in $\mathcal{B}$.
}\end{definition}

Moreover, let $B_{r}$ be the open ball in $\mathcal{B}$ with the
radius $r$ and centered at $0$ and $\partial B_{r}$ denote its boundary.
We obtain the existence and multiplicity of homoclinic solutions
of \eqref{DS} by use of the following well-known Mountain Pass Theorems,
 see \cite{Rab2}.

\begin{lemma}[{\cite[Theorem 2.2]{Rab2}}] \label{lem 2.4}
Let $\mathcal{B}$ be a real Banach space and
$I\in C^1(\mathcal{B},\mathbb{R})$ satisfying the (PS) condition.
Suppose that $I(0)=0$ and
\begin{itemize}
  \item [(A1)] there exist constants $\rho$, $\alpha>0$ such that
  $I|_{\partial B_{\rho}}\geq \alpha$,
  \item [(A2)] there exists
 $e\in {\mathcal{B}}\setminus \overline{B}_{\rho}$ such that $I(e)\leq 0$.
\end{itemize}
Then $I$ possesses a critical value $c\geq \alpha$ given by
$$
c=\inf_{g\in\Gamma}\max_{s\in[0,1]}I(g(s)),
$$
where
$$
\Gamma=\{g\in C([0,1], {\mathcal{B}}): g(0)=0,g(1)=e\}.
$$
\end{lemma}


\begin{lemma}[{\cite[Theorem 9.12]{Rab2}}] \label{lem 2.6}
Let $\mathcal{B}$ be an infinite dimensional real Banach space
and $I\in C^1(\mathcal{B},\mathbb{R})$ be even
satisfying the (PS) condition and $I(0)=0$. If
${\mathcal{B}}=V\oplus X$, where $V$ is  finite dimensional,
and $I$ satisfies
\begin{itemize}
\item [(A3)] there exist constants $\rho$, $\alpha>0$ such that
 $I|_{\partial B_{\rho}\cap X}\geq \alpha$ and

\item [(A4)] for each finite dimensional subspace
 $\tilde{\mathcal{B}}\subset \mathcal{B}$, there is an
 $R=R(\tilde{\mathcal{B}})$ such that $I\leq 0$ on
  $\tilde{\mathcal{B}}\backslash B_{R(\tilde{\mathcal{B}})}$,

\end{itemize}
 then $I$ has an unbounded sequence of critical values.
\end{lemma}


\section{Proof of Theorem \ref{Thm 1.1}}

Now we  establish the corresponding variational framework to
obtain homoclinic solutions of \eqref{DS}.
Take ${\mathcal{B}}=E$ and define the functional $I:E\to \mathbb{R}$ by
\begin{equation}\label{3.1}
\begin{aligned}
I(q)&= \int_{\mathbb{R}}\Big[\frac{1}{2}|\dot q(t)|^2
+\frac{1}{2}\big(A q(t),\dot q(t)\big)
+\frac{1}{2}\big(L(t)q(t),q(t)\big)-W(t,q(t))\Big]dt\\
&= \frac{1}{2}\|q\|^2+\frac{1}{2}\int_{\mathbb{R}}\big(A
q(t),\dot q(t)\big)dt-\int_{\mathbb{R}}W(t,q(t))dt.
\end{aligned}
 \end{equation}

\begin{lemma}\label{lem 3.0}
 Under the conditions of Theorem \ref{Thm 1.1}, we have
\begin{equation}\label{3.2}
I'(q)v=\int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A
\dot q(t),
v(t)\big)+\big(L(t)q(t),v(t)\big)-\Big(W_{q}(t,q(t)),v(t)\Big)\Big]dt,
\end{equation}
 for all $q$, $v\in E$, which yields, using the skew-symmetry
of $A$,
\begin{equation}\label{3.3}
\begin{aligned}
I'(q)q &= \|q\|^2-\int_{\mathbb{R}}\big(A\dot q(t),q(t)\big)dt
 -\int_{\mathbb{R}}\Big(W_{q}(t,q(t)),q(t)\Big)dt\\
&= \|q\|^2+\int_{\mathbb{R}}\big(A q(t),\dot q(t)\big)dt
 -\int_{\mathbb{R}}\Big(W_{q}(t,q(t)),q(t)\Big)dt.
\end{aligned}
\end{equation}
Moreover, $I$ is a continuously Fr\'echet-differentiable functional
defined on $E$; i.e., $I\in C^1(E,\mathbb{R})$ and any critical
point of $I$ on $E$ is a classical solution of \eqref{DS} with
$q(\pm\infty)=0=\dot q(\pm\infty)$.
\end{lemma}

\begin{proof}
 We begin by showing that $I:E\to \mathbb{R}$. By
(\ref{1.2}), there exist  constants $M>0$ and $R_1>0$ such that
\begin{equation}\label{3.4}
W(t,q)\leq M |q|^2 \quad \text{for all }(t,q)\in
\mathbb{R}\times \mathbb{R}^{n},\; |q|\leq R_1.
\end{equation}
 Letting $q\in E$, then $q\in C^0(\mathbb{R},\mathbb{R}^n)$
(see, e.g., \cite{Rab3}), the space of
continuous functions $q$ on $\mathbb{R}$ such that $q(t)\to 0$ as
$|t|\to +\infty$; i.e., $E\subset C^0(\mathbb{R},\mathbb{R}^n)$.
Therefore there is a constant $R_2> 0$ such that $|t|\geq R_2$
implies that $|q(t)|\leq R_1$. Hence, by (\ref{3.4}),
we have
\begin{equation}\label{3.5}
0\leq\int_{\mathbb{R}}W(t,q(t))dt\leq\int_{-R_2}^{R_2}W(t,q(t))dt+M
\int_{|t|\geq R_2}|q(t)|^2dt<+\infty.
\end{equation}
 Combining (\ref{3.1}) and (\ref{3.5}), we show that $I:E\to \mathbb{R}$.

Next we prove that $I\in C^1(E,\mathbb{R})$. Rewrite $I$ as
$I=I_1-I_2$,
where
$$
I_1:=\frac{1}{2}\int_{\mathbb{R}}\Big[|\dot q(t)|^2+\big(Aq(t),\dot q(t)\big)
 +\big(L(t)q(t),q(t)\big)\Big]dt,\quad
I_2:=\int_{\mathbb{R}}W(t,q(t))dt.
$$
It is easy to check that $I_1\in C^1(E,\mathbb{R})$, and by using the
skew-symmetry of $A$, we have
\begin{equation}\label{3.6}
I_1'(q)v=\int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A\dot q(t),
v(t)\big)+\big(L(t)q(t),v(t)\big)\Big]dt.
\end{equation}
 Therefore it is sufficient to consider $I_2$. In the process we
will see that
\begin{equation}\label{3.7}
I_2'(q)v=\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt,
\end{equation}
which is defined for all $q$, $v\in E$. For any given $q\in E$,
let us define $J(q):E\to \mathbb{R}$ as following
$$
J(q)v=\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt, \quad v\in E.
$$
It is obvious that $J(q)$ is linear. Now we show that $J(q)$ is bounded.
Indeed, for any given $q\in E$,  there exists a constant $M_1>0$
such that
$\|q\|\leq M_1$ and, by (\ref{2.1}), $\|q\|_{\infty}\leq C M_1$.
According to (H3) and (H4), there is a constant $b_1>0$
(dependent on $q$) such that
$$
|W_q(t,q(t))|\leq b_1|q(t)| \quad \text{for all } t\in \mathbb{R},
$$
which  by \eqref{1.1} and the H\"{o}lder inequality yields
\begin{equation}\label{3.8}
|J(q)v|=\Big|\int_{\mathbb{R}}\big(W_q(t,q(t)),
v(t)\big)dt\Big| \leq b_1 \|q\|_2 \,\|v\|_2
\leq \frac{b_1}{\beta}\|q\|\,\|v\|.
 \end{equation}
 Moreover, for $q$ and $v\in E$,
by the Mean Value Theorem, we have
$$
\int_{\mathbb{R}}W(t,q(t)+v(t))dt-\int_{\mathbb{R}}W(t,q(t))dt
=\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t)),
v(t)\big)dt,
$$
where $h(t)\in (0,1)$. Therefore, by Lemma \ref{lem 2.2} and
the H\"{o}lder inequality, we have
\begin{equation}\label{3.9}
\begin{aligned}
&\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t)), v(t)\big)dt
 -\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt\\
&=\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t))-W_q(t,q(t)),v(t)\big)
dt\to 0
\end{aligned}
\end{equation}
as $v\to 0$. Combining (\ref{3.8}) and (\ref{3.9}), we see
that (\ref{3.7}) holds. It remains to prove that
$I_2'$ is continuous. Suppose that $q\to q_0$ in $E$ and note that
$$
I_2'(q)v-I_2'(q_0)v=\int_{\mathbb{R}}\big(W_q(t,q(t))-W_q(t,q_0(t)),
v(t)\big)dt.
$$
By Lemma \ref{lem 2.2} and the H\"{o}lder inequality, we obtain
$$
I_2'(q)v-I_2'(q_0)v\to0 \quad \text{as }  q\to q_0,
$$
which implies the continuity of $I_2'$ and we show that
$I\in C^1(E,\mathbb{R})$.

Lastly, we check that critical points of $I$ are classical solutions
of \eqref{DS} satisfying $q(t)\to 0$ and
$\dot q(t)\to 0$ as $|t|\to +\infty$. It is well known that
$E\subset C^0(\mathbb{R},\mathbb{R}^n)$ (the space of continuous
functions $q$ on $\mathbb{R}$
such that $q(t)\to 0$ as $|t|\to +\infty$). On the other hand,
if $q$ is a critical point of $I$, for any
$v\in E\subset C^0(\mathbb{R},\mathbb{R}^n)$, by (\ref{3.2}) we have
\begin{align*}
\int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A \dot q(t),
v(t)\big)\Big]dt
&= \int_{\mathbb{R}}\big(\dot q(t)+Aq(t),\dot v(t)\big)dt\\
&= -\int_{\mathbb{R}}\big(L(t)q(t)-W_{q}(t,q(t)),v(t)\big)dt,
\end{align*}
which implies that $L(t) q-W_{q}(t,q)$ is the weak derivative of
$\dot q +A q$. Since $L\in C(\mathbb{R},\mathbb{R}^{n^2})$,
$W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})$ and
$E\subset C^0(\mathbb{R},\mathbb{R}^n)$, we see
that  $\dot q +A q$ is continuous, which yields that $\dot q$
is continuous and $q\in C^2(\mathbb{R},\mathbb{R}^n)$; i.e., $q$
is a classical solution of \eqref{DS}.
Moreover, it is easy to check that $q$ satisfies $\dot q(t)\to 0$
as $|t|\to +\infty$ since $\dot q$ is continuous.
\end{proof}


\begin{lemma}\label{lem 3.1}
Under Assumption {\rm (H1)-(H5)}, $I$ satisfies the (PS) condition.
\end{lemma}

\begin{proof}
Assume that $\{u_{j}\}_{j\in \mathbb{N}} \subset E$ is a sequence such
that $\{I(u_{j})\}_{j\in \mathbb{N}}$  is bounded and $I'(u_{j})\to 0$
as $j\to +\infty$. Then there exists a constant $C_1>0$ such that
\begin{equation}\label{3.10}
|I(u_{j})|\leq C_1,\quad
\|I'(u_{j})\|_{E^{*}}\leq C_1
\end{equation}
for every $j\in \mathbb{N}$.

We firstly prove that $\{u_{j}\}_{j\in \mathbb{N}}$ is bounded in $E$.
By (\ref{3.1}), (\ref{3.3}), (H2) and the H\"{o}lder inequality, we have
\begin{equation}\label{3.11}
\begin{aligned}
\big(\frac{\mu}{2}-1\big)\|u_j\|^2
&= \mu I(u_{j})-I'(u_{j}) u_j\\
&\quad + \int_{\mathbb{R}}\big(\mu W(t,u_j(t))
  -\big(W_{q}(t,u_j(t)),u_j(t)\Big)\big)dt\\
&\quad - \big(\frac{\mu}{2}-1\big)\int_{\mathbb{R}}\big(A u_j(t),\dot u_j(t)\big)dt\\
&\leq \mu I(u_{j})-I'(u_{j}) u_j+\big(\frac{\mu}{2}-1\big)\frac{\|A\|}{\sqrt{\beta}}\|u_j\|^2.
\end{aligned}
\end{equation}
Combining this inequality with (\ref{3.10}), we obtain
\begin{equation}\label{3.12}
\big(\frac{\mu}{2}-1\big)\big(1-\frac{\|A\|}{\sqrt{\beta}}\big)\|u_j\|^2
\leq \mu I(u_{j})-I'(u_{j}) u_j
\leq \mu C_1+ C_1 \|u_{j}\|.
\end{equation}
Since $\mu>2$ and $\|A\|<\sqrt{\beta}$, the inequality (\ref{3.12})
shows that
$\{u_{j}\}_{j\in \mathbb{N}}$ is bounded in $E$. By Lemma \ref{lem 2.1}, the
sequence $\{u_{j}\}_{j\in \mathbb{N}}$ has a subsequence, again denoted by
$\{u_{j}\}_{j\in \mathbb{N}}$, and there exists $u\in E$ such
that
\begin{gather*}
 u_{j}\rightharpoonup u,  \quad \text{weakly in } E,\\
 u_{j}\to u,\quad \text{strongly in }  L^2(\mathbb{R},\mathbb{R}^{n}).
\end{gather*}
Hence
$$
\big(I'(u_{j})-I'(u)\big) (u_{j}-u)\to 0,
$$
and by Lemma \ref{lem 2.2} and the H\"{o}lder inequality, we have
$$
\int_{\mathbb{R}}\Big(W_{q}(t,u_{j}(t))-W_{q}(t,u(t)),u_{j}(t)-u(t)\Big)dt\to 0,
$$
and
$$
\big|\int_{\mathbb{R}}\big(A \dot u_j(t)-A\dot u(t),
u_j(t)-u(t)\big)dt\big|\leq \|A\|\|\dot u_j-\dot u\|\|u_j-u\|_2\to
0
$$
as $j\to+\infty$. On the other hand, an easy computation shows that
\begin{align*}
&\big(I'(u_{j})-I'(u),u_{j}-u\big)\\
&= \|u_{j}-u\|^2
-\int_{\mathbb{R}}\big(A \dot u_j(t)-A\dot u(t), u_j(t)-u(t)\big)dt\\
&\quad -\int_{\mathbb{R}}\Big(W_{q}(t,u_{j}(t))-W_{q}(t,u(t)),u_{j}(t)-u(t)\Big)dt.
\end{align*}
Consequently, $\|u_{j}-u\| \to 0$ as $j\to +\infty$.
\end{proof}

 Now we can give the proof of Theorem \ref{Thm 1.1},
we divide the proof into several steps.

\subsection*{Proof of Theorem \ref{Thm 1.1}} \quad

\noindent\textbf{Step 1} It is clear that $I(0)=0$ by
Remark \ref{rem 1.1} and $I\in C^1(E,\mathbb{R})$ satisfies the (PS)
condition by Lemmas \ref{lem 3.0} and \ref{lem 3.1}.

\noindent \textbf{Step 2}
We now show that there exist constants $\rho>0$ and $\alpha>0$ such
that $I$ satisfies the condition (A1) of Lemma \ref{lem 2.4}.
By (\ref{1.2}), for all $\varepsilon>0$, there exists $\delta>0$
such that $W(t,q)\leq \varepsilon |q|^2$ whenever $|q|\leq \delta$.
Letting $\rho=\frac{\delta}{C}$ and $\|q\|=\rho$, we have
$\|q\|_{\infty}\leq\delta$, where $C>0$ is defined in (\ref{2.1}).
Hence $W(t,q(t))\leq \varepsilon |q(t)|^2$ for all
$t\in \mathbb{R}$. Integrating on $\mathbb{R}$, we get
$$
\int_{\mathbb{R}}W(t,q(t))dt\leq \varepsilon \|q\|_2^2\leq
\frac{\varepsilon}{\beta} \|q\|^2.
$$
In consequence, combining this with (\ref{3.1}),
we obtain that, for $\|q\|=\rho$,
\begin{equation}\label{3.13}
\begin{aligned}
I(q)&= \frac{1}{2}\|q\|^2+\frac{1}{2}\int_{\mathbb{R}}\big(A q(t),
 \dot q(t)\big)dt-\int_{\mathbb{R}}W(t,q(t))dt\\
&\geq \frac{1}{2}\|q\|^2-\frac{1}{2}\frac{\|A\|}{\sqrt{\beta}}\|q\|^2
 -\frac{\varepsilon}{\beta} \|q\|^2\\
&= \big(\frac{1}{2}-\frac{1}{2}\frac{\|A\|}{\sqrt{\beta}}
 -\frac{\varepsilon}{\beta}\big)\|q\|^2.
\end{aligned}
\end{equation}
 Setting $\varepsilon=\frac{1}{4\beta}(1-\frac{\|A\|}{\sqrt{\beta}})$,
the inequality (\ref{3.13}) implies
$$
I|_{\partial B_\rho}\geq\frac{1}{4}
\big(1-\frac{\|A\|}{\sqrt{\beta}}\big)\frac{\delta^2}{C^2}=\alpha>0.
$$

\noindent \textbf{Step 3} It remains to prove that there exists
$e\in E$ such that $\|e\|>\rho$ and
$I(e)\leq 0$, where $\rho$ is defined Step 2.
By (\ref{3.1}), we have, for every $m\in \mathbb{R}\setminus \{0\}$
 and $q\in E\setminus\{0\}$,
\begin{align*}
I(m\, q)&=  \frac{m^2}{2}\|q\|^2 +\frac{m^2}{2}
 \int_{\mathbb{R}}\big(A q(t), \dot q(t)\big)dt-\int_{\mathbb{R}}W(t,m \,q(t))dt\\
&\leq \frac{m^2}{2}\big(1+\frac{\|A\|}{\sqrt{\beta}}\big)-\int_{\mathbb{R}}W(t,m
\,q(t))dt.
\end{align*}
Take some $Q\in E$ such that $\|Q\|=1$. Then there exists a subset
$\Omega$ of positive measure of $\mathbb{R}$ such that $Q(t)\neq 0$ for
$t\in \Omega$. Take $m>0$ such that $m |Q(t)|\geq 1$ for $t\in
\Omega$. Then, by (H5) and Remark \ref{rem 2.1}, we obtain that
\begin{equation}\label{3.14}
I(m\, Q)\leq \frac{m^2}{2}\big(1+\frac{\|A\|}{\sqrt{\beta}}\big)
-m^{\mu}\int_{\Omega}\alpha_0(t) |Q(t)|^{\mu}dt.
\end{equation}
 Since $\alpha_0(t)>0$ and $\mu>2$, (\ref{3.14}) implies that
$I(m Q)<0$ for some $m>0$ such that $m |Q(t)|\geq 1$ for
$t\in \Omega$ and  $\|m Q\|> \rho$, where $\rho$ is defined in Step 2.
 By Lemma \ref{lem 2.4}, $I$ possesses a critical value
$c\geq\alpha>0$  given by
$$
c=\inf_{g\in\Gamma}\max_{s\in[0,1]}I(g(s)),
$$
where
$$
\Gamma=\{g\in C([0,1], E): g(0)=0,\,g(1)=e\}.
$$
Hence there is $q\in E$ such that
$I(q)=c$,  $I'(q)=0$.
\smallskip

\noindent \textbf{Step 4} Now suppose that $W(t,q)$ is even in $q$;
 i.e., (H6) holds, which implies that $I$ is even.
Furthermore, we already know that $I(0)=0$ and $I\in C^1(E,\mathbb{R})$
satisfies the (PS) condition in Step 1.

To apply Lemma \ref{lem 2.6}, it suffices to prove that $I$ satisfies
the conditions (A3)  and (A4) of Lemma \ref{lem 2.6}. Here we
take $V=\{0\}$ and $X=E$. (A3) is identically the same as in Step 2,
so it is already proved. Now  we prove that (A4) holds.
Let $\tilde{E}\subset E$ be a finite dimensional subspace.
 From Step 3 we know that, for any $Q\in \tilde{E}\subset E$ such
that  $\|Q\|=1$, there is $m_Q>0$ such that
$$
I(m Q)<0 \quad \text{for every } |m|\geq m_Q>0.
$$
Since $\tilde{E}\subset E$ is a finite dimensional subspace,
we can choose an $R=R(\tilde{E})>0$ such that
$$
I(q)<0, \quad \forall q\in \tilde{E}\backslash B_{R}.
$$
Hence, by Lemma \ref{lem 2.6}, $I$ possesses
an unbounded sequence of critical values $\{c_j\}_{j\in \mathbb{N}}$ with
$c_j\to +\infty$. Let $q_j$ be the critical point of $I$
corresponding to $c_j$, then \eqref{DS} has infinitely many
distinct homoclinic solutions.


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\end{document}