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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 111, 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 2014 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2014/111\hfil Existence and multiplicity]
{Existence and multiplicity of homoclinic solutions for
 $p(t)$-Laplacian systems with subquadratic potentials}

\author[B. Qin, P. Chen \hfil EJDE-2014/111\hfilneg]
{Bin Qin, Peng Chen}  % in alphabetical order

\address{Bin Qin \newline
College of  Science, China Three Gorges University,
Yichang, Hubei 443002, China}
\email{1070751409@qq.com}

\address{Peng Chen (corresponding author)\newline
College of  Science, China Three  Gorges University, Yichang, Hubei 443002, China}
\email{pengchen729@sina.com}

\thanks{Submitted December 17, 2013. Published April 16, 2014.}
\subjclass[2000]{34C37, 58E05, 70H05}
\keywords{Homoclinic solutions;  $p(t)$-Laplacian systems; genus}

\begin{abstract}
 By using the genus properties, we establish some criteria
 for the second-order $p(t)$-Laplacian  system
 $$
 \frac{d}{dt}\big(|\dot{u}(t)|^{p(t)-2}\dot{u}(t)\big)-a(t)|u(t)|^{p(t)-2}u(t)
 +\nabla W(t, u(t))=0
 $$
 to have at least one, and infinitely many   homoclinic orbits.
 where $t\in {\mathbb{R}}$, $u\in {\mathbb{R}}^{N}$,
 $p(t)\in C(\mathbb{R},\mathbb{R})$ and  $p(t)>1$,
 $a\in C({\mathbb{R}}, {\mathbb{R}})$ and
 $W\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$
 may not be periodic in $t$.
\end{abstract}

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

\section{Introduction}

 Consider the second-order ordinary $p(t)$-Laplacian system
 \begin{equation} \label{e1.1}
 \frac{d}{dt}\big(|\dot{u}(t)|^{p(t)-2}\dot{u}(t)\big)
 -a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t, u(t))=0,
 \end{equation}
 where $p\in C(\mathbb{R},\mathbb{R})$ and
 $p(t)>1$, $t\in {\mathbb{R}}$, $u\in {\mathbb{R}}^{N}$, 
 $a: {\mathbb{R}}\to {\mathbb{R}}$
 and $W: {\mathbb{R}}\times {\mathbb{R}}^{N}\to {\mathbb{R}}$. 
 As usual, we say that a solution
 $u(t)$ of \eqref{e1.1} is homoclinic (to 0) if $u(t)\to 0$ as $t\to \pm \infty$. 
 In addition, if $u(t)\not\equiv 0$ then $u(t)$ is called a nontrivial 
 homoclinic solution.


 System \eqref{e1.1} has been studied by Fan, et al. in a series of papers 
\cite{f1,f2,f3,f4}.
The $p(t)$-Laplacian systems can be applied to describe the physical
phenomena with ``pointwise different properties'' which first arose
from the nonlinear elasticity theory (see \cite{z3}). The $p(t)$-Laplacian
operator possesses more complicated nonlinearity than that of the
$p$-Laplacian, for example, it is not homogeneous, this causes many
troubles, and some classic theories and methods, such as the theory
of Sobolev spaces, are not applicable.

   It is well-known that homoclinic orbits play an important role in analyzing 
the chaos of dynamical systems.  If a system has the transversely intersected 
homoclinic orbits, then it must be chaotic.
  Therefore, it is of practical importance and mathematical significance to 
consider the existence  of homoclinic orbits of \eqref{e1.1} emanating from 0.

  If $p(t)\equiv p$ is a constant, system \eqref{e1.1} reduces to the ordinary 
$p$-Laplacian system
 \begin{equation} \label{e1.2}
   \frac{d}{dt}\big(|\dot{u}(t)|^{p-2}\dot{u}(t)\big)-a(t)|u(t)|^{p-2}u(t)
+\nabla W(t,   u(t))=0.
 \end{equation}


   In recent years, the existence and multiplicity of homoclinic orbits for 
Hamiltonian systems have been investigated in many papers via variational 
methods and many results were obtained based on
 various hypotheses on the potential functions when $p=2$, see, e.g., 
\cite{c1,c2,c3,f5,i1,r1,r2,s1,t1,z1,z2}.

   In the last decade there has been an increasing interest in the study of ordinary
 differential systems driven by the $p$-Laplacian 
(or the generalization of Laplacian \cite{m2}).
 For the existence of solutions for $p(t)$-Laplacian
Dirichlet problems on a bounded domain we refer to \cite{d1,d2,d3,d4,d5,w1}.
The study on the 
existence of solutions for $p(t)$-Laplacian equations in $\mathbb{R}$ is a new topic, 
which seems not to have been considered in the
 literature. We know that in the study
of $p$-Laplacian equations in $\mathbb{R}$, a main difficulty arises from the lack 
of compactness.
On the other hand, compared with the literature available for $W(t,x)$ being 
superquadratic  as $|x|\to +\infty$, there is less literature available for the 
case where $W(t, x)$ is subquadratic  at infinity. Motivated by papers 
\cite{c1,z1}, 
we will use the genus properties to  establish some existence criteria to 
guarantee that system \eqref{e1.1} has  infinitely many homoclinic solutions 
under more relaxed assumptions on  $W(t, x)$.

For our results, we use the following assumptions:
\begin{itemize}
\item[(A1)]  $a\in C(\mathbb{R}, (0,\infty))$ and $a(t)\to +\infty$ as $|t|\to \infty$,
$b(t)=1/a(t)$,
$b(t)^{\frac{\alpha_i(t)}{p(t)}}$ belongs to $L^{r_i(t)}(\mathbb{R},\mathbb{R})$, where
$r_i(t)$ satisfies
\[
\frac{1}{r_i(t)}+\frac{\alpha_i(t)}{p(t)}=1,\ \ i=1,2.
\]

\item[(P1)] $1<p^-:=\inf_{t\in \mathbb{R}}p(t)\leq \sup_{t\in \mathbb{R}}p(t):=p^+<\infty$;

\item[(W1)]  $W\in C^1(\mathbb{R}\times {{\mathbb{R}}}^N, {\mathbb{R}})$, $W(t,0)=0$ and there exist
 two bounded   functions $a_i(t)$  $(i=1,2.)$ such that
 $$
   |\nabla W(t, x)|\le a_1(t)\alpha_1{(t)}|x|^{\alpha_1(t)-1}, \quad \forall 
 (t, x)\in \mathbb{R}\times{\mathbb{R}}^{N}, \; |x|\le 1,
 $$
  and  for every $(t, x)\in \mathbb{R}\times{\mathbb{R}}^{N}$ with $|x|\ge 1$,
 $$
   |\nabla W(t, x)|\le a_2(t)\alpha_2{(t)}|x|^{\alpha_2(t)-1}, \quad
 |W(t, x)|\le ca_2(t){(t)}|x|^{\alpha_2(t)}
 $$
where $\alpha_i(t)$ satisfy   $\alpha_i^+<p^-$, 
$a_i(t)\in C(\mathbb{R},\mathbb{R}^+)$ $(i=1,2)$, and $c$ is a constant;

\item[(W2)]  There exist an open set $J\subset \mathbb{R} $ and  a function $\gamma_1(t)$ 
  such that
 $$
   W(t, x)\ge \eta|x|^{\gamma_1(t)}, \quad \forall 
 (t, x)\in J\times{\mathbb{R}}^{N}, \; |x|\le   1,
 $$
where $\gamma_1(t)$ satisfy $ 1<\gamma_1^+<p^-$,  $\eta>0$ is a constant;

\item[(W3)] $W(t, -x)= W(t, x)$ for all
$(t, x)\in {\mathbb{R}}\times {\mathbb{R}}^{N}$.

\end{itemize}

 Our main results are the following two theorems.

\begin{theorem} \label{thm1.1}
Assume {\rm (A1), (P1), (W1), (W2)} are satisfied.
Then  \eqref{e1.1} possesses at least one nontrivial homoclinic solution.
\end{theorem}

 \begin{theorem} \label{thm1.2}
Assume {\rm (A1), (P1), (W1), (W2), (S3)} are satisfied.
Then  \eqref{e1.1} possesses infinitely many nontrivial homoclinic solutions.
\end{theorem}

   The rest of the this article is organized as follows. 
In Section 2, we introduce some notations, preliminary
 results in space $W_a^{1, p(t)} $ and establish the corresponding 
variational structure. In Section 3, we complete the
 proofs of Theorems \ref{thm1.1}--\ref{thm1.2}. 
In Section 4, we give some  examples to to illustrate our results.

\section{Preliminaries}

    In this section, we recall some known results in critical point theory and 
the properties of space  $W_a^{1,p(t)}$ are listed for the convenience of readers.
Let $ \Omega$ be an open subset of $\mathbb{R}$. 
Let $S=\{$u$|$u is a measurable function in $\Omega\}$,
elements in $S$ that are equal to each other almost everywhere are
considered as one element.
 Define
$$
   L_a^{p(t)}(\Omega,\mathbb{R}^N)=\big\{u\in S(\Omega,\mathbb{R}^N): 
\int_\Omega a(t)|u(t)|^{p(t)}dt<\infty\big\}
$$
 with the norm
$$
 |u|_{p(t),a}=\inf\big\{\lambda>0:\int_\Omega a(t)|\frac{u}{\lambda}|^{p(t)}dt
\leq1 \big\}.
$$
Define
$$
W_a^{1,p(t)}(\Omega, \mathbb{R}^N)=\{u\in L_a^{p(t)}(\Omega,
\mathbb{R}^N):\dot{u}\in L^{p(t)}(\Omega, \mathbb{R}^N)\}
$$
with the norm
$$
 \|u\|=\inf\big\{\lambda>0:\int_\Omega \big(|\frac{\dot{u}}{\lambda}|^{p(t)}
 + a(t)|\frac{u}{\lambda}|^{p(t)}\big)dt\leq1\big\}.
 $$
We call the space $L_a^{p(t)}$ a generalized Lebesgue space, it is a
special kind of generalized Orlicz spaces. The space $W_a^{1,p(t)}$
is called a generalized Sobolev space, it is a special kind of
generalized Orlicz-Sobolev spaces. For the general theory of
generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see
\cite{a1,m4}. One can find the basic theory of spaces $L_a^{p(t)}$ and
$W_a^{1,p(t)}$ in \cite{f1,f2,f3,f4}. 

\begin{lemma}[\cite{f2,f3}] \label{lem2.1}
 Let
$$
\rho(u)=\int_\Omega a(t)|u|^{p(t)}dt,\quad \forall u\in L_a^{p(t)},
$$
then  
\begin{itemize}
\item[(i)]  $|u|_{p(t),a}<1$ $(=1;>1)$  if and only if 
$\rho(u)<1$ $(=1;>1)$;

\item[(ii)]
$|u|_{p(t),a}>1$ implies $|u|_{p(t),a}^{p^-}\leq\rho(u)\leq|u|_{p(t),a}^{p^+}$,\\
$|u|_{p(t),a}<1$ implies $|u|_{p(t),a}^{p^+}\leq\rho(u)\leq|u|_{p(t),a}^{p^-}$;


\item[(iii)] 
$|u|_{p(t),a}\to0$ if and only if $\rho(u)\to0$;\\
$|u|_{p(t),a}\to\infty$ if and only if $\rho(u)\to\infty$.

\item[(iv)]  Let $u\in L_a^{p(t)}\setminus \{0\}$, then 
$\|u\|_{p(t),a}=\lambda$ if and only if
$\rho(\frac{u}{\lambda})=1$.
\end{itemize}
\end{lemma}

 \begin{lemma}[\cite{f2,f3}] \label{lem2.2} 
Let
  $$
\varphi(u)=\int_\Omega(|\dot{u}|^{p(t)}+a(t)|u|^{p(t)})dt,\quad  \forall
 u\in W_a^{1,p(t)},
$$
\begin{itemize}
 
\item[(i)] $\|u\|<1$ $(=1;>1)$ if and only if $\varphi(u)<1\ (=1;>1)$;

\item[(ii)] $\|u\|>1$ implies $\|u\|^{p^-}\leq\varphi(u)\leq\|u\|^{p^+}$,\\
$\|u\|<1$ implies $\|u\|^{p^+}\leq\varphi(u)\leq\|u\|^{p^-}$;


\item[(iii)]  $\|u\|\to0$ if and only if $\varphi(u)\to0$;\\
 $\|u\|\to\infty$ if and only if $\varphi(u)\to\infty$.
\end{itemize}
\end{lemma}

 \begin{lemma}[\cite{f2}] \label{lem2.3}
 Let $\rho(u)=\int_\Omega a(t)|u|^{p(t)}dt$ for $u,u_n\in L_a^{p(t)}$
$(n=1,2,\cdots )$,
 then the following statements are equivalent to each other 
\begin{itemize}
\item[(i)]  $\lim_{n\to\infty}|u_n-u|_{p(t),a}=0$; 

 \item[(ii)]  $\lim_{n\to\infty}\rho(u_n-u)=0$; 

\item[(iii)]  $u_n\to u$ a.e. $t\in\Omega$ and 
$\lim_{n\to\infty}\rho(u_n)=\rho(u)$. 
\end{itemize}
\end{lemma} 


\begin{lemma}[\cite{f2}] \label{lem2.4}
 If
$\frac{1}{p(t)}+\frac{1}{q(t)}=1$,
then 
\begin{itemize}
\item[(i)]  $(L^{p(t)})^*=L^{q(t)}$, where $(L^{p(t)})^*$ is the conjugate
space of $L^{p(t)}$; 

\item[(ii)] for all $u\in L^{p(t)}$ and all $v\in L^{q(t)}$, we have
 $$
\big|\int_\Omega u(t)v(t)dt\big|\leq 2|u|_{p(t)}|v|_{q(t)}.
 $$
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{f1}] \label{lem2.5}
 If $\frac{1}{p(t)}+\frac{1}{q(t)}+\frac{1}{r(t)}=1$ and for any $u\in
L^{p(t)}(\mathbb{R},\mathbb{R})$, $v\in L^{q(t)}(\mathbb{R},\mathbb{R})$ and $w\in L^{r(t)}(\mathbb{R},\mathbb{R}) $, 
we have
\[
\int_{\mathbb{R}}|uvw|dt\leq 3|u|_{p(t)}|v|_{q(t)}|w|_{r(t)}.
\]
\end{lemma}

\begin{lemma}[\cite{f1}] \label{lem2.6}
 If $|u|^{q(x)}\in L^{s(x)/q(x)}$,
where $q,s\in L_+^\infty(\Omega), q(x)\leq s(x)$, then 
$u\in L^{s(x)}(\Omega)$ and there is a number $\overline{q}\in [q^-,q^+]$
such that $|u|^{q(x)}|_{s(x)/q(x)}=(|u|_{s(x)})^{\overline{q}}$.
\end{lemma}

\begin{lemma}[\cite{m1}] \label{lem2.7}
 If $a^{\alpha(t)/p(t)}|u|^{\alpha(t)}\in
L^{p(t)/\alpha(t)}$, then $u\in L_a^{p(t)}(\mathbb{R},\mathbb{R})$ and \\
$|a^{\alpha(t)/p(t)}|u|^{\alpha(t)}|_{p(t)/\alpha(t)}
=|u|_{p(t),a}^{\widetilde{\alpha}}$,
where $\alpha, p$ satisfy the condition {\rm (P1)} and
 $\alpha(t)<p(t)$ for all $t\in \mathbb{R}$, $\widetilde{\alpha} \in[\alpha^-, \alpha^+]$ 
is a constant.
\end{lemma}

Now, we establish the variational structure of system \eqref{e1.1}.
    Define
    $$
E=W_a^{1,p(t)}(\mathbb{R},\mathbb{R}^N)=\{u\in L_a^{p(t)}(\mathbb{R},
\mathbb{R}^N)|\dot{u}\in L^{p(t)}(\mathbb{R}, \mathbb{R}^N)\}.
$$
Let $I: E\to {\mathbb{R}}$ be defined by
 \begin{equation} \label{e2.1}
   I(u)=\int_{\mathbb{R}}\frac{1}{p(t)}(|\dot{u}|^{p(t)}+a(t)|u|^{p(t)})dt 
-\int_{\mathbb{R}}W(t,   u(t))dt.
 \end{equation}
  For convenience, we denote
\begin{equation} \label{e2.2}
 J(u)=\int_{\mathbb{R}}\frac{1}{p(t)}(|\dot{u}|^{p(t)}+a(t)|u|^{p(t)})dt, \quad
 F(u)=\int_{\mathbb{R}}W(t, u(t))dt.
  \end{equation}

\begin{lemma}[\cite{f2}] \label{lem2.8}
\begin{itemize}
\item[(i)]  $J\in C^1(E,\mathbb{R})$, and
$$
\langle J'(u), v\rangle =
\int_{\mathbb{R}}\Big(|\dot{u}(t)|^{p(t)-2}(\dot{u}(t), \dot{v}(t))
+a(t)|u(t)|^{p(t)-2}(u(t), v(t))\Big)dt,
$$
for all $u,v\in E$;

\item[(ii)] $J': E\to E^*$ is a mapping of type $(S_+)$,
i.e., if $u_n\rightharpoonup u$ and 
$$
\limsup_{n\to \infty}(J'(u_n),u_n-u)\leq 0,
$$
 then $u_n$ has a convergent subsequence in $E$. 
\end{itemize}
If  {\rm (A1), (W1)} or {\rm (W2)} hold,  then $I\in C^1(E,{\mathbb{R}})$
 and one can easily check that
 \begin{equation} \label{e2.3}
\begin{aligned}
  \langle I'(u), v\rangle 
&= \int_{\mathbb{R}}\big[|\dot{u}(t)|^{p(t)-2}(\dot{u}(t), \dot{v}(t))
                            +a(t)|u(t)|^{p(t)-2}(u(t), v(t))\\
&\quad -(\nabla W(t, u(t)), v(t))\big]dt.
\end{aligned}
 \end{equation}
Furthermore, the critical points of $I$ in $E$ are classical
solutions of \eqref{e1.1} with $u(\pm \infty)=0$.
\end{lemma}

\begin{lemma}[\cite{m1}] \label{lem2.9} 
 For $u\in  E$, then $u\in C(\mathbb{R},\mathbb{R}^N)$, and $u(t)\to 0,\ |t|\to \infty$.
  Furthermore, the embedding $E\hookrightarrow L^{\infty}(\mathbb{R},\mathbb{R}^N)$ is 
continuous and compact.
\end{lemma}

\begin{remark} \label{rmk2.1}\rm
 By Lemma \ref{lem2.9}, there exists a constant  $C>0$ such that
\begin{equation} \label{e2.4}
\|u\|_{L^\infty}\leq C\|u\|_E.
\end{equation}
\end{remark}

 \begin{lemma}[\cite{m3}] \label{lem2.10}
 Let $E$ be a real Banach space and $I\in C^1(E, {\mathbb{R}})$ satisfy the (PS)-condition. 
If $I$ is bounded from below, then $c=\inf_{E}I$ is a critical value of $I$.
\end{lemma}

To find nontrivial critical points of $I$, we will use  the ``genus''
 properties, so we recall  the following definitions and results 
(see \cite{m3}).
Let $E$ be a Banach space, $f\in C^{1}(E, \mathbb{R})$ and $c\in \mathbb{R}$. We set
\begin{gather*}
   \Sigma =\{ A\subset E-\{0\} : A \text{ is closed in $E$ and symmetric with 
respect to } 0\}, \\
   K_c=\{u\in E : f(u)=c, \ f'(u)=0\},\ \ \ \ f^c=\{u\in E : f(u)\le c\}.
\end{gather*}

\begin{definition}[\cite{m3}] \label{def2.1} 
 For $A\in \Sigma$, we say genus of $A$ is $n$ (denoted by $\gamma(A)=n$) if
 there is an odd map $\phi\in C(A, {\mathbb{R}}^n\setminus \{0\})$ and $n$ is the 
smallest integer with this property.
\end{definition}

 \begin{lemma}[\cite{m3}] \label{lem2.11}
 Let $f$ be an even $C^1$ functional on $E$ and satisfy the (PS)-condition.
 For any $n\in \mathbb{N}$, set
 $$
   \Sigma_n=\{A\in \Sigma : \gamma(A)\ge n\}, \quad
 c_n=\inf_{A\in \Sigma_n}\sup_{u\in A} f(u).
 $$
 (i) If $\Sigma_n\ne \emptyset$ and $c_n\in \mathbb{R}$, then $c_n$ is a critical
 value of $f$;

 (ii)  If there exists $r\in \mathbb{N}$ such that
 $$
   c_n=c_{n+1}=\cdots=c_{n+r}=c\in \mathbb{R},
 $$
 and $c\ne f(0)$, then $\gamma (K_c)\ge r+1$.
\end{lemma}


 \section{Proof of main results}


\begin{proof}[Proof of Theorem \ref{thm1.1}] 
 In view of Lemma \ref{lem2.8} and (W1),  $I\in C^1(E, \mathbb{R})$. In what follows, 
we first show that $I$ is coercive.
By (W1), we have
\begin{gather}
| W(t,x)|\leq a_1(t)|x|^{\alpha_1(t)},\quad |x|\leq 1, \label{e3.1}\\
| W(t,x)|\leq ca_2(t)|x|^{\alpha_2(t)},\quad |x|>1. \label{e3.2}
\end{gather}
 Assume that $\|u\| \geq 1$, by (W1), Lemma \ref{lem2.2} and Lemma 
\ref{lem2.7}, we have
 \begin{equation} \label{e3.3}
\begin{aligned}
    I(u)&  =   \int_{\mathbb{R}}\frac{1}{p(t)}(|\dot{u}|^{p(t)}+a(t)|u|^{p(t)})dt
-\int_{\mathbb{R}}W(t, u(t))dt \\
        & \geq   \frac{1}{p^+}\|u\|^{p^-}-\int_{\{t:|u(t)|\le 1\}}W(t, u(t))dt
-\int_{\{t:|u(t)|> 1\}}W(t, u(t))dt \\
        & \geq \frac{1}{p^+}\|u\|^{p^-}-\int_{\{t:|u(t)|\le 1\}}a_1(t)
|u(t)|^{\alpha_1(t)}dt-\int_{\{t:|u(t)|> 1\}}a_2(t)|u(t)|^{\alpha_2(t)}dt \\
        & \geq  \frac{1}{p^+}\|u\|^{p^-}-C_1\int_{\{t:|u(t)|\le 1\}}
b^{\alpha_1(t)/p(t)}a^{\alpha_1(t)/p(t)}|u(t)|^{\alpha_1(t)}dt \\
        & \quad -cC_2\int_{\{t:|u(t)|> 1\}}b^{\frac{\alpha_2{(t)}}{p(t)}}
a^{\frac{\alpha_2{(t)}}{p(t)}}|u(t)|^{\alpha_2(t)}dt \\
        & \geq \frac{1}{p^+}\|u\|^{p^-}-2C_1|b^{\frac{\alpha_{1}(t)}{p(t)}}
|_{L^{r_{1}(t)}}|u|_{p(t),a}^{\widetilde{\alpha_1}}
         -2cC_2|b^{\frac{\alpha_{2}(t)}{p(t)}}|_{L^{r_{2}(t)}}
|u|_{p(t),a}^{\widetilde{\alpha_2}} \\
        & \geq \frac{1}{p^+}\|u\|^{p^-}-2C_1|b^{\frac{\alpha_{1}(t)}{p(t)}}
|_{L^{r_{1}(t)}}\|u\|^{\widetilde{\alpha_1}}
         -2cC_2|b^{\frac{\alpha_{2}(t)}{p(t)}}|_{L^{r_{2}(t)}}
\|u\|^{\widetilde{\alpha_2}}.
        \end{aligned}
\end{equation}
Where $C_i=\sup_{t\in \mathbb{R}} a_i(t), \alpha_i(t), r_i(t)$ satisfy
$\frac{1}{r_i(t)}+\frac{\alpha_i(t)}{p(t)}=1,
\widetilde{\alpha_i}\in[\alpha_i^-, \alpha_i^+]$ is a constant,
$(i=1,2)$.
 By (W1), $\alpha_i^-<\alpha_i^+<p^-$,  this implies that 
$ \widetilde{\alpha_i}<p^-$. By (A), we have  $I(u)\to +\infty$ as 
$\|u\|\to +\infty$.
 Consequently, $I$ is bounded from below.


   Next, we prove that $I$ satisfies the (PS)-condition.
Assume that $\{u_k\}_{k\in {\mathbb{N}}}\subset E$
 is a sequence such that $\{I(u_k)\}_{k\in {\mathbb{N}}}$ is bounded and 
$I'(u_k)\to 0$ as $k\to +\infty$.
 Then by \eqref{e2.1} and \eqref{e3.3}, there exists a constant $A>0$ such that
 \begin{equation} \label{e3.4}
   \|u_k\|\le A, \quad k\in \mathbb{N}.
 \end{equation}
So passing to a subsequence if necessary, it can be assumed that
$u_k\rightharpoonup u$ in $E$.

By (A), $b^{\alpha_i(t)/p(t)} \in L^{r_i(t)}$ $(i=1,2)$, for
any $\varepsilon>0$, there exists $R>0$ such that
 \begin{equation} \label{e3.5}
|b(t)^{\alpha_i(t)/p(t)}|_{L^{r_i(t)}(\Omega_2)}<\varepsilon,
 \end{equation}
where $\Omega_1=\{t\in \mathbb{R}: |t|\leq R\}$, 
$\Omega_2=\mathbb{R}\setminus \Omega_1$, by Lemma \ref{lem2.9}, if $u_k\rightharpoonup
 u_0$, then $u_k\to u$ in $L^\infty$, hence, we have
\begin{gather}
\int_{\Omega_1}|W(t,u_k)-W(t,u)|dt<\varepsilon,\quad k\to
\infty, \label{e3.6}\\
\int_{\Omega_1}|\nabla W(t,u_k)-\nabla W(t,u)|dt<\varepsilon,\quad
k\to \infty. \label{e3.7}
\end{gather}
 Without loss of generality, suppose
that $\max\{\|u_k\|,\|u\|\}\leq 1$, it follows from (A1),
\eqref{e3.5}-\eqref{e3.7}, Lemma \ref{lem2.4} and Lemma \ref{lem2.7} that
\begin{equation} \label{e3.8}
\begin{aligned}
&|F(u_k)-F(u)|\\
&=\big|\int_{\mathbb{R}}(W(t, u_k(t))-W(t, u(t)))dt\big| \\
&\leq \int_{\Omega_1}| W(t, u_k(t))-W(t, u(t))|dt
 + \int_{\Omega_2}| W(t, u_k(t))-W(t, u(t))|dt \\
&\leq \varepsilon+\int_{\mathbb{R}}[| W(t, u_k(t))|+|W(t, u(t))|]dt \\
&\leq \varepsilon+\varepsilon\int_{\mathbb{R}}a_1(t)(|u_k|^{\alpha_1(t)}
 +|u|^{\alpha_1(t)})dt 
 +c\varepsilon\int_{\mathbb{R}}a_2(t)(|u_k|^{\alpha_2(t)}+|u|^{\alpha_2(t)})dt \\
&\leq \varepsilon+2C_1|b(t)^{\alpha_1(t)/p(t)}|_{L^{r_1(t)}}
|a^{\alpha_1(t)/p(t)}|u_k(t)|^{\alpha_1(t)}|_{L^{p(t)/alpha_1(t)}} \\
& \quad+2C_1|b(t)^{\alpha_1(t)/p(t)}|_{L^{r_1(t)}}
 |a^{\alpha_1(t)/p(t)}|u(t)|^{\alpha_1(t)}|_{L^{p(t)/alpha_1(t)}} \\
&\quad +2cC_2|b(t)^{\frac{\alpha_2(t)}{p(t)}}|_{L^{r_2(t)}}
 | a^{\frac{\alpha_2(t)}{p(t)}}|u_k(t)|^{\alpha_2(t)}|_{L^{p(t)/\alpha_2(t)}} \\
 & \quad+2cC_2|b(t)^{\frac{\alpha_2(t)}{p(t)}}|_{L^{r_2(t)}}
|a^{\frac{\alpha_2(t)}{p(t)}}|u(t)|^{\alpha_2(t)}|_{L^{p(t)/\alpha_2(t)}} \\
&\leq \varepsilon+ 2C_1\varepsilon(|u_k|_{L_a^{p(t)}}
 ^{\widetilde{\alpha_{1,1}}}+|u|_{L_a^{p(t)}}^{\widetilde{\alpha_{1,2}}})
 +2cC_2\varepsilon|u_k|_{L_a^{p(t)}}^{\widetilde{\alpha_{2,1}}}
 +|u|_{L_a^{p(t)}}^{\widetilde{\alpha_{2,2}}}) \\
&\leq \varepsilon+4C_1\varepsilon+4cC_2\varepsilon,
\end{aligned}
\end{equation}
where $C_i=\sup_{t\in \mathbb{R}} a_i(t)(i=1,2)$,
$ \widetilde{\alpha_{1,1}},
\widetilde{\alpha_{1,2}}\in [\alpha_1^-, \alpha_1^+],
\widetilde{\alpha_{2,1}}, \widetilde{\alpha_{2,2}}\in [\alpha_2^-,
\alpha_2^+]$.
 Hence, there exists a constant $C'$ such that
$|F(u_k)-F(u)|<C'\varepsilon$, this implies that $F(u_k)\to
F(u)$, $k\to \infty$.

On the other hand, for any $v\in E$ with $\|v\|=1$, by (W1), 
Lemmas \ref{lem2.5} and \ref{lem2.7}, we have
\begin{align*} % \albel{e3.9}
&|(F'(u_k)-F'(u),v)|\\
&\leq \int_{\Omega_1}|\nabla W(t, u_k(t))-\nabla W(t, u(t)))||v|dt \\
&\quad+\int_{\Omega_2}|\nabla W(t, u_k(t))-\nabla W(t, u(t)))||v|dt \\
&\leq  \varepsilon\|v\|_{L^\infty}+\int_{\mathbb{R}}(|\nabla W(t, u_k(t))|
 +|\nabla W(t, u(t))|)|v|dt \\
&\leq  C\varepsilon+\int_{\mathbb{R}}\alpha_1(t)a_1(t)(|u_k|^{\alpha_1(t)-1}
 +|u|^{\alpha_1(t)-1})|v|dt \\
&\quad+\int_{\mathbb{R}}\alpha_2(t)a_2(t)(|u_k|^{\alpha_2(t)-1}+|u|^{\alpha_2(t)-1})|v|dt \\
&\leq  C\varepsilon+C_3\int_{\mathbb{R}}(|u_k|^{\alpha_1(t)-1}+|u|^{\alpha_1(t)-1})|v|dt 
  +C_4\int_{\mathbb{R}}(|u_k|^{\alpha_2(t)-1}+|u|^{\alpha_2(t)-1})|v|dt \\
&= C\varepsilon+C_3\int_{\mathbb{R}}b^{\alpha_1(t)/p(t)}a^{\frac{\alpha_1(t)-1}{p(t)}}
 |u_k|^{\alpha_1(t)-1}a^{\frac{1}{p(t)}}|v|dt \\
&\quad+C_3\int_{\mathbb{R}}b^{\alpha_1(t)/p(t)}a^{\frac{\alpha_1(t)-1}{p(t)}}
 |u|^{\alpha_1(t)-1}a^{\frac{1}{p(t)}}|v|dt \\
&\quad +C_4\int_{\mathbb{R}}b^{\frac{\alpha_2(t)}{p(t)}}a^{\frac{\alpha_2(t)-1}{p(t)}}
 |u_k|^{\alpha_2(t)-1}a^{\frac{1}{p(t)}}|v|dt \\
&\quad   +C_4\int_{\mathbb{R}}b^{\frac{\alpha_2(t)}{p(t)}}a^{\frac{\alpha_2(t)-1}{p(t)}}
|u|^{\alpha_2(t)-1}a^{\frac{1}{p(t)}}|v|dt \\
&\leq  C\varepsilon +3C_3|b(t)|_{L^{r_1(t)}}^{\alpha_1(t)/p(t)}
 |a^{\frac{\alpha_1(t)-1}{p(t)}}|u_k|^{\alpha_1(t)-1}|
_{L^{\frac{p(t)}{\alpha_1(t)-1}}}
|a^{\frac{1}{p(t)}}v|_{L^{p(t)}} \\
&\quad +3C_3|b(t)|_{L^{r_1(t)}}^{\alpha_1(t)/p(t)}
 |a^{\frac{\alpha_1(t)-1}{p(t)}}|u|^{\alpha_1(t)-1}|_{L^{\frac{p(t)}{\alpha_1(t)-1}}}
 |a^{\frac{1}{p(t)}}v|_{L^{p(t)}} \\
&\quad +3C_4|b(t)|_{L^{r_2(t)}}^{\frac{\alpha_2(t)}{p(t)}}
|a^{\frac{\alpha_2(t)-1}{p(t)}}|u_k|^{\alpha_2(t)-1}
 |_{L^{\frac{p(t)}{\alpha_2(t)-1}}}
 |a^{\frac{1}{p(t)}}v|_{L^{p(t)}} \\
&\quad +3C_4|b(t)|_{L^{r_2(t)}}^{\frac{\alpha_2(t)}{p(t)}}
|a^{\frac{\alpha_2(t)-1}{p(t)}}|u|^{\alpha_2(t)-1}|_{L^{\frac{p(t)}{\alpha_2(t)-1}}}
|a^{\frac{1}{p(t)}}v|_{L^{p(t)}} \\
&\leq C\varepsilon+3C_3\varepsilon(|u_k|_{L_a^{p(t)}}
 ^{\widetilde{\alpha_{1,3}}-1}+|u|_{L_a^{p(t)}}^{\widetilde{\alpha_{1,4}}-1})
 |v|_{L_a^{p(t)}} 
 +3C_4\varepsilon(|u_k|_{L_a^{p(t)}}^{\widetilde{\alpha_{2, 3}}-1}
 +|u|_{L_a^{p(t)}}^{\widetilde{\alpha_{2, 4}}-1})|v|_{L_a^{p(t)}} \\
&\leq C\varepsilon+6C_3\varepsilon+6C_4\varepsilon.
\end{align*}
Where $C$ is defined in \eqref{e2.4}, 
$C_3=\sup_{t\in \mathbb{R}}\alpha_1(t)a_1(t)$,
$C_4=\sup_{t\in \mathbb{R}}\alpha_2(t)a_2(t)$, 
$\widetilde{\alpha_{1,3}},
\widetilde{\alpha_{1,4}}\in [\alpha_1^-, \alpha_1^+],
\widetilde{\alpha_{2,3}}, \widetilde{\alpha_{2,4}}\in [\alpha_2^-,
\alpha_2^+]$.
 Hence, there exists a constant $C''$ such that
$|F'(u_k)-F'(u)|<C''\varepsilon$, this implies that
$F'(u_k)\to F'(u)$, $k\to \infty$. This implies
that $(J'(u_k), u_k-u)\to 0$. By Lemma \ref{lem2.8}, $J'$ is a
mapping type $(S_+)$, hence, $u_k\to u$. So, $I$ satisfies
(PS) condition.

By Lemma \ref{lem2.11}, $c=\inf_E I(u)$ is a critical value of $I$, that is there
 exists a  critical point $u^*\in E$ such that $I(u^*)=c$.

   Finally, we show that $u^*\ne 0$. Let 
$u_0\in (W_0^{1, p(t)}(J)\bigcap E)\setminus \{0\}$ and
   $\|u_0\|= 1$, then by (W2) and Lemma 2.2, we have
 \begin{align} 
I(su_0)
 &  =\int_{\mathbb{R}}\frac{1}{p(t)}((|s\dot{u}|^{p(t)}+a(t)|su|^{p(t)}))dt
-\int_{\mathbb{R}}W(t, u(t))dt nonumber\\
        &  \leq \frac{s^{p^-}}{p^-}-\int_{J}W(t, su_0(t))dt nonumber\\
        & \leq \frac{s^{p^-}}{p^-}-\eta s^{\gamma_1^+}\int_{J}
|u_0(t)|^{\gamma_1(t)}dt, \quad 0<s<1. \label{e3.10}
 \end{align}
 Since $1<\gamma_1^+<p^-$, it follows from \eqref{e3.10} that $I(su_0)<0$ 
for $s>0$ small enough.
 Hence $I(u^*)=c<0$, therefore $u^*$ is nontrivial critical point of $I$, 
and so $u^*=u^*(t)$ is a
 nontrivial homoclinic solution of \eqref{e1.1}. The proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.2}]
By (W3), $I$ is an even functional. Denote
 by $\gamma(A)$ the genus of $A$.
 Set
\begin{gather*}
\Sigma =\{ A\subset E-\{0\} : A \text{ is closed in $E$ and symmetric with respect 
to } 0\},\\
\Sigma_k=\{A\in \Sigma: \gamma(A)\geq k\},\quad k=1,2,\dots,\\
c_k=\inf_{A\in \Sigma_k}\sup_{u\in A}I(u),\quad k=1,2,\dots,
\end{gather*}
we have
\begin{eqnarray*}
-\infty<c_1\leq c_2\leq \dots\leq c_k\leq c_{k+1}\leq\dots.
\end{eqnarray*}
Now let us prove that $c_k<0$ for every $k$.

 By (W2), there exists a bounded  open set $J\subset \mathbb{R}$ such that 
$W(t,x)\geq\eta|x|^{\gamma_1(t)}$, for all $t\in  J$. Since 
$W_0^{1,p(t)}(J)\subset E$, For any $k$, we can choose a $k$-dimensional
linear subspace  $E_k\subset W_0^{1,p(t)}(J)$. Since all norms of a
finite dimensional normed space are equivalent, there exists
$\rho_k\in (0,1)$ such that $u\in E_k$ with $\|u\|\leq \rho_k$
implies $|u|_{L^\infty}\leq 1$.
Set
 \[
S_{\rho_k}^{(k)}=\{u\in E_k:\|u\|=\rho_k\},
\]
for any $u\in S_{\rho_k}^{(k)}, s\in (0,1)$, we have
 \begin{align*}
I(su) &  =   \int_{J}\frac{1}{p(t)}[|s\dot{u}|^{p(t)}+a(t)|su|^{p(t)}]dt
 -\int_{J}W(t, su(t))dt\\
      & \leq 
      \frac{s^{p^-}}{p^-}\rho_k^{p^-}-s^{\gamma_1^+}\int_J|u|^{\gamma_1(t)}dt\\
      & \leq \frac{s^{p^-}}{p^-}\rho_k^{p^-}-d_ks^{\gamma_1^+}.
\end{align*}
Where $d_k=\int_J|u|^{\gamma_1(t)}dt$,  since $\gamma_1^+<p^-$,
there exist $s_k\in (0,1), \varepsilon_k>0$ such that
$$
I(s_ku)\leq -\varepsilon_k<0,\quad \forall u\in S_{\rho_k}^{(k)}.
$$
We know that $\gamma(S_{s_k\rho_k}^{(k)})=k$, so 
$c_k\leq -\varepsilon_k<0$.

By genus theory \cite{r3} and Lemma \ref{lem2.11}, each $c_k$ is a critical
value of $I$, hence there is a sequence of solutions 
$\{\pm u_k: k=1,2,\dots,\}$ of system \eqref{e1.1} such that 
$I(\pm u_k)=c_k<0$. By the
arbitraries of $k$, we can conclude that system \eqref{e1.1} have infinitely many
homoclinic solutions.
The proof is complete.
\end{proof}


\section{An example}
 In this section, we give an example to illustrate our results.
 Consider the second-order ordinary
$p(t)$-Laplacian system
 \begin{equation} \label{e4.1}
   \frac{d}{dt}\Big(|\dot{u}(t)|^{8+10|\sin t|}\dot{u}(t)\Big)
-a(t)|u(t)|^{{8+10|\sin t|}}u(t)+\nabla W(t, u(t))=0,
 \end{equation}
where $p(t)=10+10|\sin t|, a(t)=\big(1+t^2\big)^{4}$,  let
 $$
   W(t, x)=\frac{|x|^{4|\sin t|+4}}{1+t^2}+\frac{|x|^{2|\sin t|+2}}{1+t^2},
 $$
 then
\begin{gather*}
   \nabla W(t, x)=\frac{(4|\sin t|+4)|x|^{4|\sin t|+2}x}{1+t^2}
 +\frac{(2|\sin t|+2)|x|^{2|\sin t|}x}{1+t^2}, 
\\
   |\nabla W(t, x)|\le \frac{3}{1+t^2}(2+2|\sin t|)|x|^{2|\sin t|+1}, \quad
 \forall  (t, x)\in \mathbb{R}\times{\mathbb{R}}^{N}, \; |x|\le 1,
\\
    |\nabla W(t, x)|\le \frac{3}{2(1+t^2)}(4+4|\sin t|)|x|^{4|\sin t|+3}, \quad
\forall  (t, x)\in \mathbb{R}\times{\mathbb{R}}^{N}, \; |x|\ge 1,
\end{gather*}
  Let $J=(-2,2)$, $\gamma_1(t)=2|\sin t|+2$ and
 $$
   W(t, x)\ge \frac{1}{5}|x|^{2|\sin t|+2}, \quad \forall 
 (t, x)\in J\times{\mathbb{R}}^{N}, \; |x|\le 1.
 $$
 These inequalitires show that all conditions of Theorem \ref{thm1.2} are satisfied,
 where
\begin{gather*}
   \alpha_1(t)=2+2|\sin t|, \quad  \alpha_2(t)=4+4|\sin t|,\\
a_1(t)=\frac{3}{1+t^2},\quad  a_2(t)=\frac{3}{2(1+t^2)},\quad
c=\frac{4}{3}, \\
r_1(t)=\frac{5}{4},\quad r_2(t)=\frac{5}{3}.
\end{gather*}
 By Theorem \ref{thm1.2}, system \eqref{e1.1} has infinitely many 
nontrivial homoclinic solutions.

\subsection*{Acknowledgements}
The authors would like to thank the referee for his/her valuable comments 
and suggestions. This work is partially supported by the NSFC 
(No:11301297, 11261020) of China,
Scientific Research Foundation for talents of
China Three  Gorges University (KJ2012B078),
 Foundation of Hubei Educational Committee (Q20131308) and Scientific 
and Technological Innovation Projects of China Three  Gorges University.

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\end{document}