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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 29, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/29\hfil Homoclinic solutions]
{Homoclinic solutions for second-order non-autonomous
Hamiltonian systems without global Ambrosetti-Rabinowitz
conditions}

\author[R. Yuan, Z. Zhang\hfil EJDE-2010/29\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 January 14, 2010. Published February 25, 2010.}
\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 studies the existence of homoclinic solutions
 for the  second-order non-autonomous Hamiltonian system
 $$
 \ddot q-L(t)q+W_{q}(t,q)=0,
 $$
 where $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive
 definite matrix for all $t\in \mathbb{R}$.
 The function $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$
 is not assumed to  satisfy the global Ambrosetti-Rabinowitz
 condition. Assuming reasonable conditions on $L$ and $W$,
 we prove the existence of at least one nontrivial homoclinic solution,
 and for $W(t,q)$ even in $q$, we prove the existence of infinitely
 many homoclinic solutions.
\end{abstract}

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

\section{Introduction}

The purpose of this work is to study the existence of homoclinic
solutions for the  second-order
non-autonomous Hamiltonian system
\begin{equation}
\ddot q-L(t) q+W_{q}(t,q)=0, \label{HS}
\end{equation}
where $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and
definite matrix for all $t\in \mathbb{R}$,
$W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$.
We say that a solution
$q(t)$ of \eqref{HS} is homoclinic (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 $q(t)\not\equiv0$, $q(t)$ is called a nontrivial homoclinic solution.

The existence of homoclinic solutions is one of the most important
problems in the history of Hamiltonian systems, and has been
studied intensively by many mathematicians. Assuming that $L(t)$
and $W(t,q)$ are independent of $t$, or $T$-periodic in $t$, many
authors have studied the existence of homoclinic solutions for
Hamiltonian system \eqref{HS} via critical point theory and
variational methods, see for instance
\cite{Am,Caldiroli,Co2,Di2,Fl,Paturel,Rab3} and the references
therein. More general cases are considered in the recent papers
\cite{Iz,TX}. In this case, the existence of homoclinic solutions
can be obtained by going to the limit of periodic solutions of
approximating problems.


In this paper we are interested in the case where \eqref{HS}  is
non-autonomous. If $L(t)$ and $W(t,q)$ are neither autonomous nor
periodic in $t$, this problem is quite different from the ones
just described, because of the lack of compactness of the Sobolev
embedding, see for instance \cite{Korman,Om,Rab1} and the
references therein. In \cite{Rab1}, the authors considered
\eqref{HS} without periodicity assumptions on $L$  and $W$ and
showed that \eqref{HS} possesses one homoclinic solution by using
a variant of the Mountain Pass Theorem without the (PS) condition.
In \cite{Om}, under the same assumptions of \cite{Rab1}, the
authors, by employing a new compact embedding theorem, obtained
the existence and multiplicity of homoclinic solution of
\eqref{HS}. The authors in \cite{Korman} removed the technical
coercivity condition on $L$ and proved the existence of homoclinic
solutions when $L$ and $W$ are even in $t$ by the method of
approximating the homoclinic orbits from solutions of boundary
problems. By the way, a concentration-compactness principle by
Lions \cite{Loins1,Loins2} has also been used to find one
homoclinic solution of \eqref{HS}, for example \cite{Al,Ca}.


Here, we must point that all the results in the works mentioned
above are obtained under the assumption that $W$ satisfies the
global Ambrosetti-Rabinowitz condition on $q$; i.e., there is a
constant $\theta>2$ such that, for every $t\in \mathbb{R}$ and $q\in
\mathbb{R}^n\backslash \{0\}$,
\begin{equation}\label{1.1} 0<\theta
W(t,q)\leq \big(W_q(t,q),q\big),
\end{equation}
where $(\cdot,\cdot):\mathbb{R}^{n}\times \mathbb{R}^{n}\to \mathbb{R}$
denotes the standard inner product in $\mathbb{R}^{n}$ and
$|\cdot|$ is the induced norm. More recently, in \cite{Lv}, the
authors discussed the existence of even homoclinic solutions under
a class of superquadratic conditions on $W$ which is weaker than
the global Ambrosetti-Rabinowitz condition (see Corollary 2 in
\cite{Lv}). More precisely, they assumed that $W$ and $L$ are even
in $t\in \mathbb{R}$ and satisfy:
\begin{itemize}
\item[(W1)] there exist constants $d_1>0$ and $\vartheta\geq 2$ such that
$W(t,q)\leq d_1|q|^\vartheta$
for all $t\in \mathbb{R}$ and $q\in \mathbb{R}^n$;
\item[(W2)] $W(t,q)=o(|q|^2)$ as $|q|\to 0$ uniformly with respect to $t\in \mathbb{R}$;
\item[(W3)] $W_q(t,q)\to 0$ as $|q|\to 0$ uniformly with respect to $t\in \mathbb{R}$;
\item[(W4)] there exist constants $d_2>0$ and $\delta>\vartheta-2$ and $\tau\in L^1(\mathbb{R},\mathbb{R}^+)$ such that
$$
\big(W_q(t,q),q\big)-2W(t,q)\geq d_2|q|^{\delta}-\tau(t)
$$
for all $t\in \mathbb{R}$ and $q\in \mathbb{R}^n$;
\end{itemize}
and some other reasonable assumptions on $L$  and $W$.
They also gave an example which satisfies their conditions  but do
not satisfy the global  Ambrosetti-Rabinowitz condition
(\cite[Remark 2]{Lv}). On the other hand, compared with the case
that $W(t,q)$ is superquadratic as $|q|\to +\infty$, the
literature is smaller which is available for the case that
$W(t,q)$ is subquadratic as $|q|\to +\infty$. As far as the
authors are aware, only the papers \cite{Di1,ZL,ZY} dealt with
this case.

Motivated by the works mentioned above, in this paper we give some
more general conditions on $L$  and $W$ to guarantee that \eqref{HS} has nontrivial homoclinic solutions. Explicitly, assuming that
$W(t,q)$ satisfies some superquadratic conditions which are weaker than the global Ambrosetti-Rabinowitz condition and different from in \cite{Lv},
and some other reasonable assumptions on $L$ and $W$, we give a new
result to guarantee that \eqref{HS} has at least one nontrivial homoclinic solution and infinitely many homoclinic solutions if $W(t,q)$ is even in $q$, which
generalizes and improves the previous results in the literature.

To state our main result, we state the basic hypotheses on $L$
and $W$:

\begin{itemize}
  \item [(H1)]  $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric
and positive definite  matrix for all $t\in \mathbb{R}$;
there is a  continuous function $\alpha:\mathbb{R}\to \mathbb{R}$
such that $\alpha(t)>0$ for all $t\in \mathbb{R}$ and
$\big(L(t)q,q\big)\geq \alpha(t)|q|^2$
  and $\alpha(t)\to +\infty$ as $|t|\to +\infty$.
\end{itemize}
 From this condition, we see that there is a positive constant $\beta>0$ such that
\begin{equation}\label{1.2}
\big(L(t)q,q\big)\geq \beta|q|^2 \quad \text{for all } t\in \mathbb{R} ,\;
  q\in \mathbb{R}^n.
\end{equation}
\begin{itemize}
\item[(H2)] $W(t,q)\geq 0$ for all $(t,q)\in \mathbb{R}\times \mathbb{R}^{n}$ and there exist constants $M>0$ and $R_1>0$ such that
$$
W(t,q)\leq M |q|^2 \quad \text{for all }
 (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\;|q|\leq R_1,
$$
where $2M<\beta$, with $\beta$ defined in \eqref{1.2};

 \item[(H3)] there exist $\alpha_0(t)>0$ and constants $\alpha_1>2$,
$R_2>0$ such that
 $$
W(t,q)\geq \alpha_0(t)|q|^{\alpha_1}\quad \text{for all }
 (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\; |q|\geq R_2;
 $$
\item[(H4)]there exist constants $\mu>2$ and $\alpha_2$ with
$0\leq\alpha_2<(\mu-2)/2$ such that
  $$
\mu W(t,q)-\big(W_{q}(t,q),q\big)\leq \alpha_2\big(L(t)q,q\big)\quad
\text{for all } (t,q)\in \mathbb{R}\times \mathbb{R}^{n};
  $$
\item[(H5)] $W_q(t,q)=O(|q|)$ as $|q|\to 0$ uniformly with respect
to $t\in \mathbb{R}$;

 \item[(H6)] 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}

Our main result reads as follows.

\begin{theorem}\label{Thm1.1}
Suppose that the conditions {\rm (H1)-(H6)} are satisfied, then
\eqref{HS} possesses one nontrivial homoclinic solution. Moreover,
if we assume that $W(t,q)$ is even in $q$, i.e.,
\begin{itemize}
\item[(H7)] $W(t,-q)=W(t,q)$ for all $t\in \mathbb{R}$ and
$q\in \mathbb{R}^n$,
\end{itemize}
then \eqref{HS} has infinitely many  homoclinic solutions.
\end{theorem}

\begin{remark}\label{rem1.1} \rm
Note that if \eqref{1.1} holds, so does (H4), however the reverse
is not true. Now, we give an example of $W$ which satisfies
(H2)-(H6) but does not satisfy the conditions \eqref{1.1}, (W1)
and (W2). This does not say anything about (W4).
\end{remark}

\begin{example}\label{exa1.1}\rm
\begin{equation}\label{1.3}
W(t,q)=a(t)\,|q|^2\exp(|q|^\gamma),
\end{equation}
where $\gamma>0$ is a constant and $a(t)$ is a positive, continuous,
bounded function with $\inf_{t\in \mathbb{R}} a(t)>0$.
\end{example}

Then  we have
$$
W(t,q)\leq \sup_{ t \in \mathbb{R}}a(t) \exp(R_1^\gamma)|q|^2:=M |q|^2\quad
\text{for all } (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\;
|q|\leq R_1,
$$
where $R_1>0$ is any given constant, which implies that (H2)
holds if $\sup_{t\in \mathbb{R}}a(t)$ is small enough.  Moreover,
it is easy to check that
\begin{equation} \label{1.4}
\begin{gathered}
W(t,q)\geq a(t) |q|^{2+\gamma}, \\
W_q(t,q)=2a(t)\exp(|q|^\gamma)q + \gamma a(t) \exp(|q|^\gamma) |q|^\gamma q,\\
\big(W_q(t,q),q\big)=2a(t)|q|^2 \exp(|q|^\gamma)+\gamma a(t)|q|^{\gamma+2}
\exp(|q|^\gamma).
\end{gathered}
\end{equation}
So, for any constant $\mu>2$, we have
$$
\mu W(t,q)-\big(W_q(t,q),q\big)=a(t)|q|^2\exp(|q|^\gamma)
\big(\mu-2-\gamma|q|^\gamma\big),
$$
which yields
\begin{equation}\label{1.5}
0<\mu W(t,q)-\big(W_q(t,q),q\big)\leq (\mu-2)\sup_{t\in \mathbb{R}}a(t)
\exp\big(\frac{\mu-2}{\gamma}\big) |q|^2
\end{equation}
for all $(t,q)\in \mathbb{R}\times\mathbb{R}^n$ and
$0<|q|\leq\big(\frac{\mu-2}{\gamma}\big)^{1/\gamma}$;
 i.e., \eqref{1.1} does not hold for every $t\in \mathbb{R}$ and
$q\in \mathbb{R}^n  \setminus \{0\}$; and
$$
\mu W(t,q)-\big(W_q(t,q),q\big)\leq 0 \quad \text{for all }
 (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\;
|q|\geq\big(\frac{\mu-2}{\gamma}\big)^{1/\gamma},
$$
which, combining with \eqref{1.5}, implies that, for some $\mu>2$,
if $\sup_{t\in\mathbb{R}}a(t)$ is small
enough (note that \eqref{1.2}), (H4) holds.
On the other hand, by \eqref{1.3}, we have
$$
\lim_{q\to 0}\frac{W(t,q)}{|q|^2}=a(t)\geq \inf_{t\in\mathbb{R}}a(t)>0,
$$
and, by \eqref{1.4}, we have
\begin{align*}
|W_q(t,q)|&= 2a(t)\exp(|q|^\gamma)q + \gamma a(t) \exp(|q|^\gamma) |q|^\gamma q \\
&\leq \sup_{t\in\mathbb{R}}a(t) \exp(|q|^\gamma)(2+\gamma |q|^\gamma)|q|
\end{align*}
and
$$
2\inf_{t\in \mathbb{R}}a(t)\leq\lim_{q\to 0}\frac{|W_q(t,q)|}{|q|}
=2a(t)\leq 2\sup_{t\in\mathbb{R}}a(t).
$$

The remainder of the paper is organized as follows.
In section 2, some preliminary results are given.
In section 3, we give the proof of Theorem \ref{Thm1.1}.

\section{Preliminary Results}

To establish our result via the critical point theory,
we firstly describe some properties of the space on which
the variational framework associated
with \eqref{HS} is defined. Let
$$
E=\big\{q\in H^1(\mathbb{R},\mathbb{R}^n):
\int_{\mathbb{R}}\big[|\dot q(t)|^2+\big(L(t)q(t),q(t)\big)\big]dt<+\infty\big\},
$$
then the space $E$ is a Hilbert space 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 embedding 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
$$
\|q\|_{\infty}:=\mathop{\rm ess\,sup}\{|q(t)|: t\in \mathbb{R} \}.
$$

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

Similar to Lemma 2 of \cite{Om}, we can get the following result.
For the reader's convenience, we give the details of its proof.

\begin{lemma}\label{lem 2.2}
Suppose that {\rm (H1), (H5), (H6)} are satisfied.
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{proof}
Assume that $q_k\rightharpoonup q_0$ in $E$. Then,
by Banach-Steinhaus Theorem and \eqref{2.1}, there is a constant
$b_1>0$ such that
$$
\sup_{k\in \mathbb{N}}\|q_k\|_{\infty}\leq b_1.
$$
Assumptions (H5) and (H6) imply the existence of $b_2>0$ such that
$$
|W_q(t,q_k(t))|\leq b_2 |q_k(t)|
$$
for all $k\in \mathbb{N}$ and $t\in \mathbb{R}$. Hence
$$
|W_q(t,q_k(t))-W_q(t,q_0(t))|\leq b_2(|q_k(t)|+|q_0(t)|)\leq b_2(|q_k(t)-q_0(t)|+2|q_0(t)|).
$$
Since, by Lemma \ref{lem 2.1}, $q_k\to q_0$ in $L^2$, passing to a subsequence if necessary, it can be assumed that
$$
\sum_{k=1}^{\infty}\|q_k-q_0\|_2<+\infty,
$$
which implies that $q_k(t)\to q_0(t)$ for almost every $t\in \mathbb{R}$ and
$$
\sum_{k=1}^{\infty}|q_k(t)-q_0(t)|=\zeta(t)\in L^2(\mathbb{R},\mathbb{R}^n).
$$
Therefore,
$$
|W_q(t,q_k(t))-W_q(t,q_0(t))|\leq b_2(\zeta(t)+2|q_0(t)|).
$$
Then, using the Lebesgue's Convergence Theorem, the lemma
is readily proved.
\end{proof}


Now we introduce more notations and some necessary 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$\acute{e}$chet-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{HS} 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 that
\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
 $\tilde{E}\subset \mathcal{B}$, there is an $R=R(\tilde{E})$
 such that $I\leq 0$ on
  $\tilde{E}\backslash B_{R(\tilde{E})}$.
\end{itemize}
 Then $I$ has an unbounded sequence of critical values.
\end{lemma}


\section{Proof of Theorem \ref{Thm1.1}}

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

\begin{lemma}\label{lem 3.0}
 Under the conditions of Theorem \ref{Thm1.1}, we have
\begin{equation}\label{3.2}
I'(q)v=\int_{\mathbb{R}}\big[\big(\dot q(t),\dot 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 that
\begin{equation}\label{3.3}
I'(q)q =\|q\|^2-\int_{\mathbb{R}}\big(W_{q}(t,q(t)),q(t)\big)dt.
\end{equation}
Moreover, $I$ is a continuously Fr$\acute{e}$chet-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{HS} with
$q(\pm\infty)=0=\dot q(\pm\infty)$.
\end{lemma}

\begin{proof}
We firstly show that $I:E\to \mathbb{R}$. By (H2), 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}
Let $q\in E$, then $q\in C^0(\mathbb{R},\mathbb{R}^n)$
(the space of continuous functions $q$ on $\mathbb{R}$ such
that $q(t)\to 0$ as $|t|\to +\infty$). Therefore there is a constant
$R> 0$ such that $|t|\geq R$ implies that $|q(t)|\leq R_1$.
Hence, by \eqref{3.4}, we have
\begin{equation}\label{3.5}
\int_{\mathbb{R}}W(t,q(t))dt\leq\int_{-R}^{R}W(t,q(t))dt
+M \int_{|t|\geq R}|q(t)|^2dt<+\infty.
\end{equation}
Combining \eqref{3.1} and \eqref{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
$$
I_1:=\int_{\mathbb{R}}\big[\frac{1}{2}|\dot q(t)|^2+\frac{1}{2}\big(L(t)q(t),q(t)\big)
\big]dt\quad \text{and}\quad I_2:=\int_{\mathbb{R}}W(t,q(t))dt.
$$
It is easy to check that $I_1\in C^1(E,\mathbb{R})$ and
\begin{equation}\label{3.6}
I_1'(q)v=\int_{\mathbb{R}}\big[\big(\dot q(t),\dot v(t)\big)+\big(L(t)q(t),
v(t)\big)\big]dt.
\end{equation}
Therefore it is sufficient to show this is the case for $I_2$. 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
$$
J(q)v=\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt, \quad v\in E.
$$
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 \eqref{2.1}, $\|q\|_{\infty}\leq C M_1$.
So according to (H5) and (H6), there is a constant $b_3>0$ such that
$$
|W_q(t,q(t))|\leq b_3|q(t)|, \quad \text{for all} \quad t\in \mathbb{R},
$$
which yields that, by \eqref{1.2} and the H\"{o}lder inequality,
\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_3 \|q\|_2 \|v\|_2\leq \frac{b_3}{\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 \eqref{3.8} and \eqref{3.9}, we see that \eqref{3.7} holds.
It remains to prove that
$I_2'$ is continuous. Suppose that $q\to q_0$ in $E$ and note that
\begin{align*}
 \sup_{\|v\|=1}|I_2'(q)v-I_2'(q_0)v|
&= \sup_{\|v\|=1}|\int_{\mathbb{R}}\big(W_q(t,q(t))-W_q(t,q_0(t)), v(t)\big)dt|\\
&\leq \|W_q(\cdot,q(\cdot))-W_q(\cdot,q_0(\cdot))\|_2\|v\|_2\\
&\leq \frac{\|v\|}{\sqrt{\beta}}\|W_q(\cdot,q(\cdot))
 -W_q(\cdot,q_0(\cdot))\|_2.
\end{align*}
 By Lemma \ref{lem 2.2}, we obtain
$I_2'(q)v-I_2'(q_0)v\to 0$ as $ \|q\|\to \|q_0\|$
uniformly with respect to $v$,
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{HS} satisfying $q(t)\to 0$ and
$\dot q(t)\to 0$ as $|t|\to +\infty$. We have known that
$E\subset C^0(\mathbb{R},\mathbb{R}^n)$. Moreover if $q$ is one critical
point of $I$, by \eqref{3.2}, we have $L(t) q-W_{q}(t,q)$ is
the weak derivative of $\dot q$. Since $L\in C(\mathbb{R},\mathbb{R}^{n^2})$,
$W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})$ and
$E\subset C^0(\mathbb{R},\mathbb{R}^n)$, which yields that
$q\in C^2(\mathbb{R},\mathbb{R}^n)$; i.e., $q$ is a classical
solution of \eqref{HS}. Moreover, it is easy to check that
$\dot q(t)\to 0$ as $|t|\to +\infty$.
\end{proof}


\begin{lemma}\label{lem 3.1}
Under the conditions {\rm (H1), (H4), (H5), (H6)},
$I$ 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 \eqref{3.1}, \eqref{3.3} and (H4), 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
  + \int_{\mathbb{R}}\big(\mu W(t,u_j(t))-\big(W_{q}(t,u_j(t)),u_j(t)\big)\big)dt
 \\
&\leq \mu I(u_{j})-I'(u_{j}) u_j
 +\alpha_2\int_{\mathbb{R}}\big(L(t)u_j(t),u_j(t)\big)dt.
\end{aligned}
\end{equation}
Let us define
$$
\eta(q)=\int_{\mathbb{R}}\Big[\big(\frac{\mu-2}{2}\big)|\dot q(t)|^2
+\big(\frac{\mu-2}{2}-\alpha_2\big)\big(L(t)q(t),q(t)\big)\Big]dt,
$$
then we have
\begin{equation}\label{3.12}
\mu_1\|q\|^2\leq\eta(q)\leq \mu_2\|q\|^2,
\end{equation}
where $\mu_1=\frac{\mu-2}{2}-\alpha_2$ and $\mu_2=\frac{\mu-2}{2}$.
Thus, combining \eqref{3.10}, \eqref{3.11} with \eqref{3.12},
we obtain
\begin{equation}\label{3.13}
\mu_1\|u_j\|^2
\leq\eta(u_j) \leq \mu I(u_{j})-I'(u_{j}) u_j
\leq \mu C_1+ C_1 \|u_{j}\|.
\end{equation}
Since $\mu_1>0$, the inequality \eqref{3.13} 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
 \[
\int_{\mathbb{R}}\Big(W_{q}(t,u_{j}(t))-W_{q}(t,u(t)),u_{j}(t)-u(t)\Big)dt\to 0
\]
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(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 present the proof of Theorem \ref{Thm1.1}, divided
into several steps.

\subsection*{Proof of Theorem \ref{Thm1.1}}
\textbf{Step 1.} It is clear that $I(0)=0$ by (H2)
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}. In fact, assume that
$q\in E$ and $0<\|q\|_{\infty}\leq R_1$. Then, by \eqref{1.2}
 and (H2), we have
\[
\int_{\mathbb{R}}W(t,q(t))dt\leq M \int_{\mathbb{R}} |q(t)|^2dt
\leq M \|q\|_2^2\leq \frac{M}{\beta}\|q\|^2,
\]
and in consequence, combining this with \eqref{3.1}, we obtain
\begin{equation}\label{3.14}
I(q) \geq \frac{1}{2}\|q\|^2-\frac{M}{\beta}\|q\|^2\\[0.2cm]
 = \frac{1}{2}\big(1-\frac{2M}{\beta}\big)\|q\|^2.
\end{equation}
Note that (H2) implies $1-\frac{2 M}{\beta}>0$. Set
\begin{equation}\label{3.15}
\rho=\frac{R_1}{C},\quad
\alpha=\frac{(\beta-2 M) R_1^2}{2 \beta C^2}>0.
\end{equation}
By \eqref{2.1}, if $\|q\|=\rho$, then $0<\|q\|_{\infty}\leq R_1 $
and \eqref{3.14} gives that
$I|_{\partial B_{\rho}}\geq \alpha$.


\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 in \eqref{3.15}.
By \eqref{3.1}, we have, for every $m\in \mathbb{R}\setminus \{0\}$
and $q\in E\setminus\{0\}$,
$$
I(m\, q)= \frac{m^2}{2}\|q\|^2 -\int_{\mathbb{R}}W(t,m \,q(t))dt.
$$
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 $m |Q(t)|\geq R_2$ for $t\in \Omega$.
Then, by (H2) and (H3), we obtain that
\begin{equation}\label{3.16}
I(m\, Q)\leq \frac{m^2}{2}-m^{\alpha_1}
\int_{\Omega}\alpha_0(t) |Q(t)|^{\alpha_1}dt.
\end{equation}
 Since $\alpha_0(t)>0$ and $\alpha_1>2$, \eqref{3.16} implies that
$I(m\, Q)<0$ for some $m>0$ such that $m |Q(t)|\geq R_2$ for
$t\in \Omega$ and
 $\|m\, Q\|> \rho$, where $\rho$ is defined in \eqref{3.15}.
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
$$
I(q)=c, \quad I'(q)=0.
$$

\noindent \textbf{Step 4.} Now suppose that $W(t,q)$ is even in $q$;
 i.e., (H7) holds, which implies that $I$ is even.
Moreover, 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}. (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 $\tilde{R}=R(\tilde{E})>0$ such that
$$
I(q)<0, \quad \forall q\in \tilde{E}\backslash B_{\tilde{R}}.
$$
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{HS} has infinitely many  homoclinic solutions.


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\end{document}