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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 64, pp. 1--13.\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{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/64\hfil Periodic solutions]
{Periodic solutions for non-autonomous second-order differential
systems with $(q,p)$-Laplacian}

\author[C. Li, Z.-Q. Ou, C.-L. Tang \hfil EJDE-2014/64\hfilneg]
{Chun Li, Zeng-Qi Ou,  Chun-Lei Tang}  % in alphabetical order

\address{Chun Li \newline
School of Mathematics and Statistics,
Southwest University, Chongqing 400715, China}
\email{Lch1999@swu.edu.cn}

\address{Zeng-Qi Ou \newline
School of Mathematics and Statistics,
Southwest University, Chongqing 400715, China}
\email{ouzengq707@sina.com}

\address{Chun-Lei Tang (corresponding author)\newline
School of Mathematics and Statistics,
Southwest University, Chongqing 400715, China\newline
Tel +86 23 68253135, fax +86 23 68253135}
\email{tangcl@swu.edu.cn}

\thanks{Submitted March 18, 2012. Published March 5, 2014.}
\subjclass[2000]{34C25, 35B38, 47J30}
\keywords{Periodic solution; differential systems;
$(q,p)$-Laplacian;  \hfill\break\indent
least action principle;  saddle point theorem}

\begin{abstract}
 Some existence theorems are obtained for periodic solutions of
 nonautonomous second-order differential systems with
 $(q,p)$-Laplacian by using the least action principle and the
 saddle point theorem.
\end{abstract}

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

\section{Introduction}

Consider the second-order system
\begin{equation}\label{1}
\begin{gathered}
\frac{d}{dt} \big( | \dot{u}_1(t) |^{q-2} \dot{u}_1(t)  \big)
 = \nabla_{u_1} F(t,u_1(t),u_2(t)),\\
\frac{d}{dt} ( | \dot{u}_2(t) |^{p-2} \dot{u}_2(t)  )
= \nabla_{u_2} F(t,u_1(t),u_2(t)),\quad \text{a.e. } t\in [0,T],\\
u_1(0)-u_1(T) = \dot{u}_1(0)-\dot{u}_1(T) = 0,\\
u_2(0)-u_2(T) = \dot{u}_2(0)-\dot{u}_2(T)=0,
\end{gathered}
\end{equation}
where $1<p$, $q <\infty$, $T>0$  and  $|\cdot|$  denotes the
Euclidean norm in $ \mathbb{R}^N$.
$F:[0,T]\times  \mathbb{R}^N \times  \mathbb{R}^N\to  \mathbb{R}$
satisfies the following assumption
\begin{itemize}
\item[(A1)]
\begin{itemize}
 \item $F$ is measurable in $t$ for each $(x_1, x_2)\in
 \mathbb{R}^N\times  \mathbb{R}^N$;
  \item $F$ is
continuously differentiable in $(x_1, x_2)$ for a.e. $t\in [0,T]$;
  \item there exist $a_1, a_2 \in C( \mathbb{R}_+,
 \mathbb{R}_+)$ and $b \in L^1(0,T; \mathbb{R}_+)$ such that
\[
|F(t,x_1,x_2)|, \text{ } |\nabla_{x_1}F(t,x_1,x_2)|, \text{ }
|\nabla_{x_2}F(t,x_1,x_2)| \leq \big[a_1(|x_1|) +
a_2(|x_2|)\big]b(t)
\]
for all $(x_1, x_2)\in   \mathbb{R}^N\times  \mathbb{R}^N$ and a.e.
$t\in [0,T]$.
\end{itemize}
\end{itemize}

We denote by $W_T^{1,p}$ the
Sobolev space of functions $u\in L^p(0,T; \mathbb{R}^N)$ having a
weak derivative $\dot{u} \in L^p(0,T; \mathbb{R}^N)$. The norm in
$W_T^{1,p}$ is defined by
\[
\|u \|_{W_T^{1,p}} = \Big( \int_0^T \big( |u(t)|^p +
|\dot{u}(t)|^p\big)dt \Big)^{1/p}.
\]
The corresponding functional $\varphi:W\to \mathbb{R} $ given is
\[
 \varphi(u_1, u_2)= \frac{1}{q}\int_0^T |\dot{u}_1(t) |^q dt +
\frac{1}{p}\int_0^T |\dot{u}_2(t) |^p dt + \int_0^T
F(t,u_1(t),u_2(t))dt,
\]
where $W=W_T^{1,q}\times W_T^{1,p}$  is a reflexive Banach
space and endowed
with the norm
\[
\|(u_1,u_2) \|_W = \|u_1\|_{W_T^{1,q}} +
\|u_2\|_{W_T^{1,p}}.
\]
It follows from assumption (A1) that the functional $\varphi$ is
continuously differentiable and weakly lower semicontinuous on   $ W$.
Moreover,
\begin{align*}
&\langle\varphi'(u_1,u_2),(v_1,v_2)\rangle\\
&=\int_0^{T}[( |\dot{u}_1 (t)|^{q-2}\dot{u}_1(t),\dot{v}_1(t))
+ (\nabla_{u_1} F(t,u_1(t),u_2(t)),v_1(t))]dt\\
&\quad+\int_0^{T}[( |\dot{u}_2 (t)|^{p-2}\dot{u}_2(t),\dot{v}_2(t))
+ (\nabla_{u_2} F(t,u_1(t),u_2(t)),v_2(t))]dt
\end{align*}
for all $(u_1,u_2),(v_1,v_2)\in W $.

For each $u\in W_T^{1,p}$ can be written as
$u(t)=\bar{u} + \tilde{u}(t)$ with
\[
\bar{u} = \frac{1}{T}\int_0^T u(t)dt,\quad
\int_0^T \tilde{u}(t)dt=0.
\]
We have the Sobolev's inequality (for a proof and details see \cite{mw})
\[
\| \tilde{u} \|_\infty \le C_1 \| \dot{u} \|_p,\ \| \tilde{v}
\|_\infty \le C_1 \| \dot{v} \|_q\quad \text{for each } u
\in W_T^{1,p},\ v \in W_T^{1,q},
\]
and Wirtinger's inequality (see \cite{mw})
\[
\| \tilde{u} \|_p \le C_2 \| \dot{{u}} \|_p,\quad
\| \tilde{v} \|_q \le C_2 \| \dot{{v}} \|_q\quad \text{for each } u \in
W_T^{1,p},\ v \in W_T^{1,q},
\]
where
\[
\|u\|_p=\Big(\int_0^T|u(t)|^pdt\Big)^{1/p},\quad \|u\|_\infty=\max_{t\in[0,T]}|u(t)|.
\]
A function $G:\mathbb{R}^N\to \mathbb{R}$ is called to be
$(\lambda,\mu)$-subconvex if
\[
G(\lambda(x+y))\leq \mu(G(x)+G(y))
\]
for some $\lambda, \mu>0$ and all $x,y\in \mathbb{R}^N$
(see \cite{Wu-Tang}).

The  existence of periodic solutions for the second-order Hamiltonian
system
 \begin{equation}\label{HS}
\begin{gathered} \ddot{u}(t) =\nabla F(t,u),\quad\text{a.e. }t\in[0,T],\\
u (0)-u (T) = \dot{u} (0)-\dot{u} (T) = 0,
\end{gathered}
\end{equation}
has been extensively investigated in  papers, such as \cite{Berger-Schechter,LongYM,Mawhin,mw,Tang1,Tang2,tang,Tang-Wu,Tang-Meng,Willem,Wu-Tang} and the reference therein.
Many solvability conditions are given, such as the coercive condition (see \cite{Berger-Schechter}), the periodicity condition (see \cite{Willem}), the convexity condition (see \cite{ Mawhin}), the
boundedness condition (see \cite{mw}), the subadditive condition  (see \cite{Tang1}), and
the sublinear condition (see \cite{tang}). When the gradient $\nabla F(t,x)$ is
bounded; that is, there exists $g \in L^1( 0,T;\mathbb{R}_+)$ such that
 \[
 |\nabla F(t,x)|\leq g(t)
 \]
for all $x\in  \mathbb{R}^N$ and a.e. $t\in [0,T]$.
 Mawhin and Willem  \cite{mw} obtained the existence of solutions for problem \eqref{HS} under the condition
\[
\int_0^TF(t,x)dt\to+\infty\text{(or $-\infty$)},\quad\text{as } |x|\to\infty.
\]
Tang  \cite{tang} proved the existence of solutions for problem \eqref{HS} when
\begin{equation}\label{sub}
 |\nabla F(t,x)|\leq f(t)|x|^\alpha+g(t)
\end{equation}
for all $x\in  \mathbb{R}^N$ and a.e.
$t\in [0,T]$, where $f,g \in L^1( 0,T;\mathbb{R}_+)$ and $\alpha\in[0,1)$.
 And, $F$ satisfies the condition
\[
|x|^{-2\alpha} \int_0^TF(t,x)dt\to+\infty\ (or\ -\infty),\ \text{as}\ |x|\to\infty.
\]
Tang and Meng \cite{Tang-Meng} studied the existence of solutions for problem \eqref{HS} under the conditions \eqref{sub} or
 \[
 |\nabla F(t,x)|\leq f(t)|x| +g(t)
 \]
for all $x\in  \mathbb{R}^N$ and a.e.
$t\in [0,T]$, where $f,g \in L^1(0,T;\mathbb{R}_+)$.
The results in \cite{Tang-Meng}  complement those in \cite[Theorem 1 and 2]{tang}.

Recently,    Pa\c{s}ca and Tang \cite{P-Tang} established  the existence
results for problem \eqref{1} which extend  \cite[Theorems 1 and 2]{tang}.
By applying the least action principle,
Pa\c{s}ca \cite{Pasca-Simon} proved   some existence theorems  for
problem \eqref{1}  which generalize the corresponding Theorems
of \cite{Wu-Tang}.  Using the  Saddle Point Theorem, Pa\c{s}ca and
Tang \cite{Pasca-Tang} obtained some existence results
for problem \eqref{1}.   Pa\c{s}ca \cite{Pasca-Singap} studied the existence
 of periodic solutions for  nonautonomous
second-order differential inclusions systems with ($q,p)$-Laplacian
which extend the  results of \cite{ Pasca-PanAmer,Pasca-Commun,P-Tang,tang}.

In this paper, motivated by references \cite{Pasca-Simon,P-Tang,tang,Tang-Meng},
we consider  the existence of  periodic solutions for problem \eqref{1} by
using the least action principle and the Saddle Point Theorem.
Our main results are the following theorems.

\begin{theorem}\label{Th1}
 Suppose that $F=F_1+F_2$, where $F_1$ and $F_2$
satisfy assumption {\rm (A1)} and the following conditions:
\begin{itemize}
 \item[(H0)] $F_1(t,\cdot,\cdot)$ is $(\lambda,\mu)$-subconvex
 with
$\lambda>1/2$ and $1/2<\mu<2^{r-1}\lambda^r$ for $ a.e. t\in[0,T]$,
 where $r=\min\{p,q\}$;

 \item[(H1)] there exist $f_i, g_i, h_i \in L^1(0,T; \mathbb{R}_+)$,
  $i=1,2$, $\alpha_1 \in [0,q-1)$, $\alpha_2 \in [0,p-1)$,
  $\beta_1\in [0, p/q')$, $\beta_2\in [0, q/p')$,  $q'=q/(q-1)$ and $p'=p/(p-1)$
\begin{gather*}
|\nabla_{x_1} F_2(t,x_1,x_2)| \leq f_1(t) |x_1|^{\alpha_1}
 + g_1(t) |x_2|^{\beta_1}+h_1(t)
\\
|\nabla_{x_2} F_2(t,x_1,x_2)| \leq f_2(t) |x_2|^{\alpha_2}
 + g_2(t) |x_1|^{\beta_2}+h_2(t)
\end{gather*}
for all $(x_1, x_2)\in  \mathbb{R}^N \times  \mathbb{R}^N$ and a.e.
$t\in [0,T]$;

\item[(H2)]
\[
\lim_{|x|\to \infty}\frac{1}{|x_1|^{\gamma_1} + |x_2|^{\gamma_2}}
\Big(\frac{1}{\mu}\int_0^T
F_1(t,\lambda x_1,\lambda x_2)dt+\int_0^T F_2(t,x_1,x_2)dt\Big)>2K,
\]
where  $|x|= \sqrt{|x_1|^2 + |x_2|^2} $, $\gamma_1=\max\{q'\alpha_1,\,\beta_2p' \}, \,\gamma_2=\max\{p'\alpha_2,\,\beta_1q'\}$ and
\begin{align*}
K=\max\Big\{&\frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'},\frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'},\\
   &\frac{4^{q'/q}(2^{\beta_1} \|g_1\|_{L^1}C_1)^{q'} }{q'},
   \frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'} }{p'} \Big\}.
\end{align*}
\end{itemize}
Then problem \eqref{1} has at least one solution in $W$.
\end{theorem}

\begin{corollary}\label{cor1}
Suppose that $F=F_1+F_2$, satisfies {\rm (H0), (H1)}  and
\begin{itemize}
  \item [(H2')]
\[
\frac{1}{|x_1|^{\gamma_1} + |x_2|^{\gamma_2}}\Big(\frac{1}{\mu}\int_0^T
F_1(t,\lambda x_1,\lambda x_2)dt+\int_0^T
F_2(t,x_1,x_2)dt\Big)\to  \infty
\]
\end{itemize}
as $|x| \to +\infty$. Then problem
\eqref{1} has at least one solution in $W$.
\end{corollary}

\begin{remark}\label{rem1} \rm
Corollary \ref{cor1} generalizes Theorem 1 of  \cite{Pasca-Simon}. In
fact, it follows from Corollary \ref{cor1} by letting $\beta_1=\beta_2=0$. There
are functions satisfying the assumptions of our Corollary \ref{cor1} and
not satisfying the assumptions in \cite{Pasca-Simon,P-Tang}.
For example, Let $\alpha_1=\alpha_2=15/4$, $\beta_1=\beta_2=11/4$,
$p=q=5$, $p'=q'=5/4$, and
 \begin{gather*}
 F_1(t,x_1)=5+\sin(|x_1|^6+|x_2|^6),  \\
 F_2(t,x_1,x_2)=\big(\frac{2T}{3}-t\big)(|x_1|^{19/4}
 +|x_2|^{19/4}+|x_1|^{5/4}|x_2|^{5/4}).
\end{gather*}
\end{remark}


\begin{theorem}\label{Th2}
Suppose that $F(t,x_1,x_2)$ satisfies   {\rm (H1)} and
\begin{itemize}
  \item [(H3)]
  \[
   \lim_{|x|\to \infty}\frac{1}{|x_1|^{\gamma_1} + |x_2|^{\gamma_2}} \int_0^T
F(t,x_1,x_2)dt <-( 2q'+ 2p'+1)  2K.
\]
\end{itemize}
  Then problem \eqref{1} has at least one solution in $W$.
\end{theorem}

\begin{corollary}\label{cor2}
Suppose that $F(t,x_1,x_2)$ satisfies  {\rm (H1)} and
\begin{itemize}
  \item [(H3')]
\[
\frac{1}{|x_1|^{\gamma_1} + |x_2|^{\gamma_2}} \int_0^T
F(t,x_1,x_2)dt \to -\infty
\]
\end{itemize}
as $|x| \to \infty$. Then problem
\eqref{1} has at least one solution in $W$.
\end{corollary}


\begin{remark}\label{rem2}\rm
 Corollary \ref{cor2} extends \cite[Theorem 2]{P-Tang}.
In fact, it follows from Corollary \ref{cor2} by letting
$\beta_1=\beta_2=0$.  There are functions satisfying the
assumptions of our Corollary \ref{cor2} and not satisfying the assumptions in
\cite{P-Tang}.  For example, Let
$\alpha_1=\alpha_2=15/4$, $\beta_1=\beta_2=11/4$, $p=q=5$, $p'=q'=5/4$,
and
\[
F(t,x_1,x_2)=\big(\frac{ T}{3}-t\big)(|x_1|^{19/4}
+|x_2|^{19/4}+|x_1|^{5/4}|x_2|^{5/4}).
\]
\end{remark}


\section{Proofs of main results}

 Tian and Ge \cite{tg} proved the following result  which generalizes
 a very well known result proved by
Jean Mawhin and Michel Willem  \cite[Theorem 1.4]{mw}.

\begin{lemma}[\cite{tg}] \label{lem1}
Let $L:[0,T]\times  \mathbb{R}^N \times  \mathbb{R}^N \times
 \mathbb{R}^N \times  \mathbb{R}^N \to  \mathbb{R}$,
$(t, x_1, x_2, y_1, y_2) \to L(t, x_1, x_2, y_1, y_2)$
be measurable in $t$ for each $(x_1, x_2, y_1, y_2)$, and continuously
differentiable in $(x_1, x_2, y_1, y_2)$ for a.e. $t\in [0,T]$.
If there exist $a_i \in C(\mathbb{R}_+,  \mathbb{R}_+)$, $i=1,2$,
$b\in L^1(0,T;  \mathbb{R}_+)$,
and $c_1 \in L^p(0,T;  \mathbb{R}_+)$, $c_2 \in L^q(0,T; \mathbb{R}_+)$,
$1< p, q < \infty$, such that for a.e. $t\in [0,T]$ and every
$(x_1, x_2, y_1, y_2)\in  \mathbb{R}^N \times \mathbb{R}^N \times
 \mathbb{R}^N \times  \mathbb{R}^N$, one has
\begin{gather*}
|L(t,x_1,x_2,y_1,y_2)|\leq
(a_1(|x_1|)+a_2(|x_2|))(b(t)+|y_1|^q + |y_2|^p), \\
|D_{x_1}L(t,x_1,x_2,y_1,y_2)|\leq
(a_1(|x_1|)+a_2(|x_2|))(b(t) + |y_2|^p), \\
|D_{x_2}L(t,x_1,x_2,y_1,y_2)|\leq
(a_1(|x_1|)+a_2(|x_2|))(b(t)+|y_1|^q),\\
|D_{y_1}L(t,x_1,x_2,y_1,y_2)|\leq (a_1(|x_1|) + a_2(|x_2|))(c_1(t)+|y_1|^{q-1}), \\
|D_{y_2}L(t,x_1,x_2,y_1,y_2)|\leq ( a_1(|x_1|) + a_2(|x_2|))(c_2(t)+|y_2|^{p-1}),
\end{gather*}
then the function $\varphi: W_T^{1,q}\times W_T^{1,p} \to
 \mathbb{R}$ defined by
\[
\varphi(u_1,u_2) = \int_0^T L(t,u_1(t), u_2(t), \dot{u}_1(t), \dot{u}_2(t))dt
\]
is continuously differentiable on $W_T^{1,q}\times W_T^{1,p}$ and
\begin{align*}
\langle \varphi'(u_1,u_2), (v_1,v_2)\rangle 
&= \int_0^T ((D_{x_1}L(t,u_1(t), u_2(t), \dot{u}_1(t), \dot{u}_2(t)),
v_1(t)) \\
&\quad +(D_{y_1}L(t,u_1(t), u_2(t), \dot{u}_1(t), \dot{u}_2(t)),
\dot{v}_1(t))\\
&\quad + (D_{x_2}L(t,u_1(t), u_2(t), \dot{u}_1(t), \dot{u}_2(t)),
v_2(t))\\
&\quad +(D_{y_2}L(t,u_1(t), u_2(t), \dot{u}_1(t),
\dot{u}_2(t)), \dot{v}_2(t))) dt.
\end{align*}
\end{lemma}


\begin{corollary}\label{cor3} 
Let $L:[0,T]\times  \mathbb{R}^N \times  \mathbb{R}^N \times
 \mathbb{R}^N \times  \mathbb{R}^N \to  \mathbb{R}$ be defined by
\[
L(t, x_1, x_2, y_1, y_2) = \frac{1}{q}|y_1|^q + \frac{1}{p}|y_2|^p
+ F(t, x_1, x_2)
\]
where $F:[0,T]\times  \mathbb{R}^N \times  \mathbb{R}^N \to
\mathbb{R}$ satisfies condition {\rm (A1)}. 
If $(u_1, u_2)\in W_T^{1,q}\times W_T^{1,p}$ is a solution of the corresponding
Euler equation $\varphi'(u_1,u_2)=0$, then $(u_1, u_2)$ is a
solution of problem  \eqref{1}.
\end{corollary}

\begin{remark}\label{rem3} \rm
The function $\varphi$ is weakly lower semi-continuous (w.l.s.c.)
 on $W$ as the sum of two convex continuous functions and of a
 weakly continuous one.
\end{remark}

We will prove Theorem \ref{Th1} by using the least action principle 
\cite[Theorem 1.1]{mw},  and Theorem  \ref{Th2}
by using the saddle point theorem \cite[Theorem 4.6]{rab}.

\begin{proof}[Proof of Theorem \ref{Th1}]
Let $\beta= \log_{2\lambda}(2\mu)$. Then
$0<\beta<r$. For $|x|>1$, there exists a positive integer $n$ such
that
\[
n-1<\log_{2\lambda}|x|\leq n.
\]
So, we have $|x|^\beta>(2\lambda)^{(n-1)\beta}=(2\mu)^{n-1}$ and
$|x|\leq (2\lambda)^n$. Then, by   (A1) and (H0), one has
\begin{align*}
 F_1(t,x_1,x_2)&\leq 2\mu
 F_1(t,x_1/(2\lambda),x_2/(2\lambda))\leq\dots\\
&\leq (2\mu)^nF_1
 (t,x_1/(2\lambda),x_2/(2\lambda))\\
 &\leq 2\mu|x|^\beta (a_{10}+a_{20})b(t)
\end{align*}
for a.e. $t\in [0,T] $ and all $|x|>1$, where
$a_{i0}=\max_{0\leq s\leq 1}a_i(s)$, $i=1,2$. Therefore, 
\begin{equation}\label{F1-leq}
F_1(t,x_1,x_2)\leq (2^{\beta/2+1}\mu(|x_1|^\beta+|x_2|^\beta)+1)(a_{10}+a_{20})b(t)
\end{equation}
for a.e. $t\in [0,T] $ and all $(x_1, x_2)\in  \mathbb{R}^N \times  \mathbb{R}^N$.

It follows from (H1), Sobolev's inequality and
 Young's inequality  that
\begin{align*} 
&\Big| \int_0^T  ( F_2(t,u_1(t),\bar{u}_2) -
F_2(t,\bar{u}_1,\bar{u}_2))dt \Big|   \\
&=\Big| \int_0^T \int_0^1 ( \nabla_{x_1} F_2(t,\bar{u}_1 + s
\tilde{u}_1(t),\bar{u}_2) , \tilde{u}_1(t) ) ds dt \Big|
\\
&\leq \int_0^T \int_0^1 f_1(t) |\bar{u}_1 +
s\tilde{u}_1(t)|^{\alpha_1} |\tilde{u}_1(t)| ds dt + \int_0^T
\int_0^1 g_1(t)|\bar{u}_2|^{\beta_1} |\tilde{u}_1(t)| ds dt \\
&\quad +   \int_0^T
\int_0^1 h_1(t) |\tilde{u}_1(t)| ds dt\\
&\leq 2^{q-1} ( |\bar{u}_1|^{\alpha_1} +
\|\tilde{u}_1\|_{\infty}^{\alpha_1}  ) \|\tilde{u}_1\|_{\infty}
\|f_1\|_{L^1}+|\bar{u}_2|^{\beta_1} \|\tilde{u}_1\|_{\infty} \|g_1\|_{L^1}
 +\|\tilde{u}_1\|_{\infty}\|h_1\|_{L^1}\\
&\leq  2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\| {\dot{u}}_1\|_{q}^{\alpha_1 +1} 
 +  2^{q-1}\|f_1\|_{L^1}C_1|\bar{u}_1|^{\alpha_1} \| \dot{u}_1\|_q\\
&\quad +C_1\|g_1\|_{L^1}|\bar{u}_2|^{\beta_1}\|\dot{u}_1\|_{q}
 + C_1\|h_1\|_{L^1}\|\dot{u}_1\|_{q}\\
&\leq 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_1\|_q^{\alpha_1 +1} +  \frac{1}{4q}
\|\dot{u}_1\|_q^q + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_1|^{q'\alpha_1}\\
&\quad + \frac{1}{4q}\|\dot{u}_1\|_q^q +\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'}
   }{q'} |\bar{u}_2|^{q'\beta_1}+C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q \\
&= \frac{1}{2q}
\|\dot{u}_1\|_q^q+ 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_1\|_q^{\alpha_1 +1} + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_1|^{q'\alpha_1}\\
&\quad +\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'} }{q'} 
|\bar{u}_2|^{q'\beta_1}+C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q
\end{align*}
and
\begin{align*}
 &\Big| \int_0^T ( F_2(t,u_1(t),u_2(t)) -
F_2(t,u_1(t),\bar{u}_2) )dt \Big| \\
&=   \Big| \int_0^T \int_0^1 ( \nabla_{x_2} F_2(t,u_1(t),\bar{u}_2 +
s \tilde{u}_2(t)), \tilde{u}_2(t)) ds dt \Big|\\
&\leq \int_0^T \int_0^1 f_2(t) |\bar{u}_2 +
s\tilde{u}_2(t)|^{\alpha_2} |\tilde{u}_2(t)| ds dt + \int_0^T
\int_0^1 g_2(t)| u _1  |^{\beta_2} |\tilde{u}_2(t)| ds dt \\
&\quad+ \int_0^T\int_0^1 h_2(t) |\tilde{u}_2(t)| ds dt\\
&\leq   2^{p-1}(|\bar{u}_2|^{\alpha_2}+ \|\tilde{u}_2\|_{\infty}^{\alpha_2}) 
 \|\tilde{u}_2\|_{\infty}
\|f_2\|_{L^1} +  2^{\beta_2}(|\bar{u}_1|^{\beta_2}
 + \|\tilde{u}_1\|_{\infty}^{\beta_2}) \|\tilde{u}_2\|_{\infty}
\|g_2\|_{L^1}\\
&\quad +\|\tilde{u}_2\|_{\infty} \|h_2\|_{L^1}\\
&\leq  2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1} \|\dot{u}_2\|_{p}^{\alpha_2 +1} 
+ 2^{p-1}C_1\|f_2\|_{L^1}|\bar{u}_2|^{\alpha_2} \|\dot{u}_2\|_{p}
 +  C_1\|h_2\|_{L^1}\|\dot{u}_2\|_{p}\\
&\quad +2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1}\|\dot{u}_2\|_{p}
 \|\dot{u}_1\|_{q}^{\beta_2}
+2^{\beta_2}\|g_2\|_{L^1}C_1\|\dot{u}_2\|_{p}|\bar{u } _1|^{\beta_2}\\
&\leq   2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1} 
 + \frac{1}{4p}
\|\dot{u}_2\|_p^p +\frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}}{p'}
 |\bar{u}_2|^{p'\alpha_2}\\
&\quad + \frac{1}{4p}\|\dot{u}_2\|_p^p  
 +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'} 
 \|\dot{u}_1\|_q^{\beta_2p'} +\frac{1}{4p}\|\dot{u}_2\|_p^p\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}}{p'}|\bar{u}_1|^{\beta_2p'}
+C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p\\
&= \frac{3}{4p} \|\dot{u}_2\|_p^p + 2^{p-1}C_1^{\alpha_2
+1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1} +
\frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}}{p'}|\bar{u}_2|^{p'\alpha_2}\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'} 
 \|\dot{u}_1\|_q^{\beta_2p'}+\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_1|^{\beta_2p'}\\
&\quad +C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p
\end{align*}
for all $(u_1,u_2)\in W$. So, one has
\begin{align*} 
&\Big| \int_0^T (F_2(t,u_1(t),u_2(t))-F_2(t,\bar{u}_1,\bar{u}_2)  )dt \Big|\\
&\leq \Big| \int_0^T ( F_2(t,u_1(t),\bar{u}_2)
  -F_2(t,\bar{u}_1,\bar{u}_2) )dt \Big|\\
&\quad +\Big| \int_0^T( F_2(t,u_1(t),u_2(t))-F_2(t,u_1(t),\bar{u}_2))dt \Big|\\
&\leq  \frac{1}{2q}\|\dot{u}_1\|_q^q+\frac{3}{4p}
\|\dot{u}_2\|_p^p+C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q+C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p\\
&\quad + 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_1\|_q^{\alpha_1 +1}
 + 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1}\\
&\quad + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}}{q'}|\bar{u}_1|^{q'\alpha_1}
+\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'}}{q'} |\bar{u}_2|^{q'\beta_1}\\
&\quad + \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_2|^{p'\alpha_2}+\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_1|^{p'\beta_2}\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'} 
\|\dot{u}_1\|_q^{\beta_2p'}
\end{align*}
for all $(u_1,u_2)\in W$.
Hence, we obtain from (H0),  \eqref{F1-leq} and the above expression that
  \begin{align*}
\varphi(u_1,u_2)
&=  \frac{1}{q} \int_0^T |\dot{u}_1(t)|^q dt +
\frac{1}{p} \int_0^T |\dot{u}_2(t)|^p dt+\int_0^T  F_1(t,u_1(t),u_2(t))dt \\
&\quad + \int_0^T(F_2(t,u_1(t),u_2(t)) - F_2(t, \bar{u}_1,\bar{u}_2)
) dt + \int_0^T F_2(t, \bar{u}_1,\bar{u}_2) dt \\
&\geq \frac{1}{2q} \|\dot{u}_1\|_q^q  + \frac{1}{4p} \|\dot{u}_2\|_p^p
 -C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q -C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p\\
&\quad - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_1\|_q^{\alpha_1 +1}
 - 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1}\\
&\quad -\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}
   }{p'} \|\dot{u}_1\|_q^{p'\beta_2}- \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_1|^{q'\alpha_1}\\
&\quad -\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'}
   }{q'} |\bar{u}_2|^{q'\beta_1}- \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_2|^{p'\alpha_2}\\
&\quad -\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_1|^{\beta_2p'}
   + \int_0^T F_2(t,\bar{u}_1,\bar{u}_2) dt\\
&\quad +\frac{1}{\mu}\int_0^T F_1(t,\lambda\bar{u}_1,\lambda\bar{u}_2)dt
 -\int_0^T F_1(t,- \tilde{ u }_1,-\tilde{ u }_2)dt\\
&\geq \frac{1}{2q} \|\dot{u}_1\|_q^q  + \frac{1}{4p} \|\dot{u}_2\|_p^p 
 -\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}
   }{p'} \|\dot{u}_1\|_q^{p'\beta_2}\\
&\quad - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_1\|_q^{\alpha_1 +1}
 - 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1}\\
&\quad -C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q -C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p\\
&-(2^{\beta/2+1}C_1^\beta\mu(\|\dot{u}_1\|_q^\beta
 +\|\dot{u}_2\|_p^\beta)+1)(a_{10}+a_{20})\int_0^Tb(t)dt\\
&\quad +( |\bar{u}_1|^{\gamma_1}+|\bar{u}_2|^{\gamma_2} ) 
  (\frac{1}{|\bar{u}_1|^{\gamma_1}+|\bar{u}_2|^{\gamma_2}} 
 (\frac{1}{\mu}\int_0^T F_1(t,\lambda\bar{u}_1,\lambda\bar{u}_2)dt\\
&\quad  + \int_0^T F_2(t,\bar{u}_1,\bar{u}_2) dt)-2K)-K_0
\end{align*}
for all $(u_1,u_2)\in W$ and some positive constants $K$ and $K_0$.
It follows  that $\varphi(u_1,u_2) \to +\infty$ as $\|(u_1,u_2)\|_W \to \infty$ 
due to (H2). By \cite[Theorem 1.1]{mw} and Corollary \ref{cor3}, The proof is
 complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Th2}]
Firstly, we prove that $\varphi$
satisfies the $(PS)$ condition. Suppose that $\{(u_{1n},u_{2n})\}$ is
a $(PS)$ sequence for $\varphi$, that is, $\varphi'(u_{1n},u_{2n})
\to 0$ as $n\to \infty$ and $\{\varphi(u_{1n},u_{2n})\}$ is
bounded. In a way similar to the proof of Theorem \ref{Th1}, we
have
\begin{align*}
&\Big| \int_0^T ( \nabla_{x_1} F(t,u_{1n}(t),u_{2n}(t)),
\tilde{u}_{1n}(t) ) dt \Big|\\
&\leq  2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1} \|\dot{u}_{1n}\|_q^{\alpha_1 +1} +
\frac{3}{4q} \|\dot{u}_{1n}\|_q^q + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{1n}|^{q'\alpha_1}\\
&\quad+\frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1^{\beta_1+1})^{q'}}{q'}
\|\dot{u}_{2n}\|^{\beta_1q'}_p + \frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{2n}|^{q'\beta_1}\\
&\quad+ C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q
\end{align*}
and
\begin{align*}
& \Big| \int_0^T ( \nabla_{x_2}
F(t,u_{1n}(t),u_{2n}(t)),
\tilde{u}_{2n}(t) ) dt \Big| \\
&\leq   2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1} \|\dot{u}_{2n}\|_p^{\alpha_2 +1} +
\frac{3}{4p} \|\dot{u}_{2n}\|_p^p + \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{2n}|^{p'\alpha_2}\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'}
\|\dot{u}_{1n}\|^{\beta_2p'}_q + \frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{1n}|^{p'\beta_2}\\
&\quad +  C_1\|h_2\|_{L^1}\|\dot{u}_{2n}\|_p
\end{align*}
for all $n$. Hence, one has
\begin{equation}\label{u-1n}
\begin{aligned}
&\|(\tilde{u}_{1n},\tilde{u}_{2n} )\|_W \\
&\geq \langle \varphi'
(u_{1n}, u_{2n}), (\tilde{u}_{1n},\tilde{u}_{2n}) \rangle\\
&=  \int_0^T ( ( \nabla_{x_1} F(t,u_{1n}(t),u_{2n}(t)),
\tilde{u}_{1n}(t) ) + ( |\dot{u}_{1n}(t)|^{q-2}
\dot{u}_{1n}(t), \dot{{u}}_{1n}(t) )\\
&+( \nabla_{x_2} F(t,u_{1n}(t),u_{2n}(t)), \tilde{u}_{2n}(t) )
+ (  |\dot{u}_{2n}(t)|^{p-2}
\dot{u}_{2n}(t), \dot{{u}}_{2n}(t) )  )dt \\
&\geq \frac{4q-3}{4q} \|\dot{u}_{1n}\|_q^q+\frac{4p-3}{4p} \|\dot{u}_{2n}\|_p^p - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1} \|\dot{u}_{1n}\|_q^{\alpha_1+1}\\
&\quad -\frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1^{\beta_1+1})^{q'}
}{q'}\|\dot{u}_{2n}\|^{\beta_1q'}_p -\frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}}{q'}|\bar{u}_{1n}|^{q'\alpha_1}\\
&\quad - \frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{2n}|^{q'\beta_1}- 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1} \|\dot{u}_{2n}\|_p^{\alpha_2 +1}\\
&\quad -\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'}
\|\dot{u}_{1n}\|^{\beta_2p'}_q - \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{2n}|^{p'\alpha_2}  \\
&\quad -\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{1n}|^{p'\beta_2}-C_1\|h_2\|_{L^1}
\|\dot{u}_{2n}\|_p- C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q
\end{aligned}
\end{equation}
for large $n$. It follows from Wirtinger's inequality that
\begin{equation}\label{W_ineq}
\begin{aligned}
 \|(\tilde{u}_{1n}, \tilde{u}_{2n})\|_W
&= \|\tilde{u}_{1n}\|_{W_T^{1,q}} + \|\tilde{u}_{2n}\|_{W_T^{1,p}}\\
&\leq  (1+C^q_2)^{1/q} \|\dot{{u}}_{1n}\|_q +
(1+C^p_2)^{1/p} \|\dot{{u}}_{2n}\|_p  \\
&\leq \max \big\{(1+C^q_2)^{1/q}, (1+C^p_2)^{1/p} \big\} \big(
 \|\dot{{u}}_{1n}\|_q + \|\dot{{u}}_{2n}\|_p \big)
\end{aligned}
\end{equation}
for all $n$. So, it follows from \eqref{u-1n}   and
\eqref{W_ineq} that
\begin{align*}
&K( |\bar{u}_{1n}|^{p'\beta_2}+|\bar{u}_{2n}|^{p'\alpha_2}+
 |\bar{u}_{1n}|^{q'\alpha_1}+|\bar{u}_{2n}|^{q'\beta_1})\\
&\geq \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{1n}|^{q'\alpha_1}+ \frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{2n}|^{q'\beta_1}\\
&\quad +\frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{2n}|^{p'\alpha_2}  +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{1n}|^{p'\beta_2}\\
&\geq   \frac{4q-3}{4q} \|\dot{u}_{1n}\|_q^q+\frac{4p-3}{4p} \|\dot{u}_{2n}\|_p^p \\
&\quad - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1} \|\dot{u}_{1n}\|_q^{\alpha_1 +1}-\frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1^{\beta_1+1})^{q'}}{q'}
\|\dot{u}_{2n}\|^{\beta_1q'}_p\\
&\quad - 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1} \|\dot{u}_{2n}\|_p^{\alpha_2 +1}-\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'}
\|\dot{u}_{1n}\|^{\beta_2p'}_q\\
&\quad -C_1\|h_2\|_{L^1}\|\dot{u}_{2n}\|_p-C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q
-\|(\tilde{u}_{1n},\tilde{u}_{2n} )\|_W\\
&\geq \frac{4q-3}{4q} \|\dot{u}_{1n}\|_q^q+\frac{4p-3}{4p} \|\dot{u}_{2n}\|_p^p-C_1\|h_2\|_{L^1}\|\dot{u}_{2n}\|_p- C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q \\
&\quad - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1} \|\dot{u}_{1n}\|_q^{\alpha_1 +1}-\frac{4^{q'/q}(2^{\beta_1}\|g_1\|_{L^1}C_1^{\beta_1+1})^{q'}}{q'}
\|\dot{u}_{2n}\|^{\beta_1q'}_p\\
&\quad - 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1} \|\dot{u}_{2n}\|_p^{\alpha_2 +1}-\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}}{p'}
\|\dot{u}_{1n}\|^{\beta_2p'}_q\\
&\quad -(1+C^p_2)^{1/p} \|\dot{{u}}_{2n}\|_p -(1+C^q_2)^{1/q} \|\dot{{u}}_{1n}\|_q\\
&\geq   \frac{ q-1}{q} \|\dot{u}_{1n}\|_q^q+\frac{ p-1}{ p} \|\dot{u}_{2n}\|_p^p
  -K_1\\
&=   \frac{ 1}{q'} \|\dot{u}_{1n}\|_q^q+\frac{ 1}{ p'} \|\dot{u}_{2n}\|_p^p-K_1
\end{align*}
for large $n$ and some positive constant $K_1$. Hence, by the above expression,
 we obtain
 \begin{equation}\label{u-1n-k1}
  2K( |\bar{u}_{1n}|^{\gamma_1}+|\bar{u}_{2n}|^{\gamma_2})
  \geq \min\big\{ \frac{ 1}{q'},\frac{ 1}{ p'}\big\}
\big(\|\dot{u}_{1n}\|_q^q+\|\dot{u}_{2n}\|_p^p\big)  -K_2
\end{equation}
 for large $n$ and some positive constant $K_2$.
 By the proof of Theorem
\ref{Th1}, we have
\begin{align*}
&\Big| \int_0^T (
F(t,u_{1n}(t),u_{2n}(t))-F(t,\bar{u}_{1n},\bar{u}_{2n}))dt \Big|\\
&\leq \frac{1}{2q}\|\dot{u}_{1n}\|_q^q+\frac{3}{4p}
\|\dot{u}_{2n}\|_p^p+C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q
+C_1\|h_2\|_{L^1}\|\dot{u}_{2n}\|_p \\
&\quad + 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_{1n}\|_q^{\alpha_1 +1}+ 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_{2n}\|_p^{\alpha_2 +1}\\
&\quad + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{1n}|^{q'\alpha_1}+\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'}
   }{q'} |\bar{u}_{2n}|^{q'\beta_1}\\
&\quad + \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{2n}|^{p'\alpha_2}+\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{1n}|^{p'\beta_2}\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}
   }{p'} \|\dot{u}_{1n}\|_q^{\beta_2p'}
\end{align*}
for all $n$. It follows from the boundedness of 
$\{\varphi(u_{1n}, u_{2n}) \}$, \eqref{u-1n-k1}  and the above inequality
that
\begin{align*}
K_3
&\leq  \varphi(u_{1n}, u_{2n})\\
&= \frac{1}{q} \int_0^T |\dot{u}_{1n}(t)|^q dt + \frac{1}{p}
\int_0^T |\dot{u}_{2n}(t)|^p dt\\
&\quad + \int_0^T \big[ F(t,u_{1n}(t),u_{2n}(t)) - F(t,
\bar{u}_{1n},\bar{u}_{2n}) \big] dt + \int_0^T F(t,
\bar{u}_{1n},\bar{u}_{2n}) dt \\
&\leq  \frac{3}{2q}
\|\dot{u}_{1n}\|_q^q+\frac{7}{4p}
\|\dot{u}_{2n}\|_p^p+C_1\|h_1\|_{L^1}\|\dot{u}_{1n}\|_q+C_1\|h_2\|_{L^1}\|\dot{u}_{2n}\|_p \\
&\quad + 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}\|\dot{u}_{1n}\|_q^{\alpha_1 +1}+ 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_{2n}\|_p^{\alpha_2 +1}\\
&\quad + \frac{4^{q'/q}(2^{q-1}\|f_1\|_{L^1}C_1)^{q'}
   }{q'}|\bar{u}_{1n}|^{q'\alpha_1}+\frac{4^{q'/q}( \|g_1\|_{L^1}C_1)^{q'}
   }{q'} |\bar{u}_{2n}|^{q'\beta_1}\\
&\quad + \frac{4^{p'/p}(2^{p-1}\|f_2\|_{L^1}C_1)^{p'}}{p'}|\bar{u}_{2n}|^{p'\alpha_2}
+\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1)^{p'}
   }{p'}|\bar{u}_{1n}|^{p'\beta_2}\\
&\quad +\frac{4^{p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}
   }{p'} \|\dot{u}_{1n}\|_q^{\beta_2p'} +\int_0^T F(t,
\bar{u}_{1n},\bar{u}_{2n}) dt\\
&\leq  2(\|\dot{u}_{1n}\|_q^q+
\|\dot{u}_{2n}\|_p^p)+ 2K( |\bar{u}_{1n}|^{\gamma_1}+|\bar{u}_{2n}|^{\gamma_2})
    +\int_0^T F(t,\bar{u}_{1n},\bar{u}_{2n}) dt+K_4\\
&\leq    (2\max \{ q',  p' \}+1 )
  2K( |\bar{u}_{1n}|^{\gamma_1}+|\bar{u}_{2n}|^{\gamma_2})
    +\int_0^T F(t,\bar{u}_{1n},\bar{u}_{2n}) dt+K_5\\
&\leq    ( 2q'+ 2p'+1 )
  2K( |\bar{u}_{1n}|^{\gamma_1}+|\bar{u}_{2n}|^{\gamma_2})
    +\int_0^T F(t,\bar{u}_{1n},\bar{u}_{2n}) dt+K_5\\
&\leq (|\bar{u}_{1n}|^{\gamma_1} + |\bar{u}_{2n}|^{\gamma_2})
 ( \frac{1}{ |\bar{u}_{1n}|^{\gamma_1} +|\bar{u}_{2n}|^{\gamma_2}} \int_0^T F(t,
\bar{u}_{1n},\bar{u}_{2n}) dt\\
&\quad +  ( 2q'+ 2p'+1 )  2K  )  + K_5
\end{align*}
for large $n$ and some real constants $K_3$, $K_4$  and $K_5$. The
above inequality and (H3) imply that 
$ (|\bar{u}_{1n}|^{\gamma_1} + |\bar{u}_{2n}|^{\gamma_2} )$ is
bounded. Hence, $(u_{1n}, u_{2n})$ is bounded by \eqref{W_ineq} and \eqref{u-1n-k1}.
 By the compactness of the embedding $W^{1,p}_{T}$( or $W^{1,q}_{T}$)
$\subset C(0,T;\mathbb{R}^N)$, the sequence $\{u_{1n}\}$
(or $\{u_{2n}\}$) has a subsequence, still denoted by $\{u_{1n}\}$
(or $\{u_{2n}\}$), such that
\begin{gather}\label {12}
u_{1n} \text{ (or $u_{2n}$) }\rightharpoonup u_1\text{ (or $u_{2}$) }
\quad \text{ weakly  in }  W_{T}^{1,p}\text{ (or in $W_{T}^{1,q}$)},\\
\label {13}
u_{1n}\text{ (or $u_{2n}$) } \to u_1\text{ (or $u_{2}$)} \quad
\text{strongly  in } C( 0,T;\mathbb{R}^N) .
\end{gather}
Note that
\begin{equation}\label{varphi}
\begin{aligned}
&\langle \varphi'(u_{1n},u_{2n}),(u_1-u_{1n},0)\rangle\\
&= \int_{0}^{T}|\dot{u}_{1n}(t)|^{p-2}(\dot{u}_{1n}(t), \dot{u}_1
 - \dot{u}_{1n}(t))dt\\
&\quad - \int_{0}^{T}(\nabla_{x_1} F(t,u_{1n}(t),u_{2n}(t)),
u_1(t)-u_{1n}(t))dt \to 0
\end{aligned}
\end{equation}
as $n\to\infty$. From
\eqref{13}, $\{u_{1n}\}$ is bounded in $C(0,T;\mathbb{R}^N)$. Then we
have
\begin{align*}
&\Big|\int_{0}^{T}(\nabla_{x_1}
F(t,u_{1n}(t),u_{2n}(t)),u_1(t)-u_{1n}(t))dt\Big|\\
&\leq \int_{0}^{T}|\nabla_{x_1} F(t,u_{1n}(t),u_{2n}(t))|\cdot |u_1(t)-u_{1n}(t)|dt \\
&\leq  K_6\int_{0}^{T}b(t)|u_1(t)-u_{1n}(t)|dt \\
&\leq  K_6\|b\|_{L^{1}} \|u_1-u_{1n}\|_{\infty}
\end{align*}
for some positive constant $K_6$, which combines with \eqref{13}
implies that
\[
\int_{0}^{T}(\nabla_{x_1} F(t,u_{1n}(t),u_{2n}(t)),u_1(t)-u_{1n}(t))dt\to 0 \quad
\text{as } n\to\infty.
\]
Hence, by \eqref{varphi}, one has
\[
\int_{0}^{T}|\dot{u}_{1n}(t)|^{p-2}(\dot{u}_{1n}(t),\dot{u}_1(t)
-\dot{u}_{1n}(t))dt \to 0 \quad \text{as }  n\to\infty \ .
\]
Moreover,  from \eqref{13} we obtain
\[
\int_{0}^{T}|u_{1n}(t)|^{p-2}(u_{1n}(t),u_1(t)-u_{1n}(t))dt\to 0 \quad
\text{as }  n\to\infty \,.
\]
Setting
\[
\psi(u_1,u_2)=\frac{1}{p}\int_{0}^{T}(|u_1(t)|^p+|\dot{u}_1(t)|^p)dt
+\frac{1}{q}\int_{0}^{T}(|u_2(t)|^q+|\dot{u}_2(t)|^q)dt,
\]
one obtains
\begin{align*}
\langle \psi'(u_{1n},u_{2n}),(u_1-u_{1n},0)\rangle
&= \int_{0}^{T}|u_{1n}(t)|^{p-2}(u_{1n}(t),u_1(t)-u_{1n}(t))dt\\
&\quad +\int_{0}^{T}|\dot{u}_{1n}(t)|^{p-2}(\dot{u}_{1n}(t),\dot{u}_1(t)
 -\dot{u}_{1n}(t))dt
\end{align*} 
and
\begin{equation}\label{14}
\langle \psi'(u_{1n},u_{2n}),(u_1-u_{1n},0)\rangle\to 0 \
\ \text{as} \ n\to\infty \,.
\end{equation}
By the H\"{o}lder's inequality, we have
\[
0\leq (\|u_{1n}\|^{p-1}-\|u_1\|^{p-1})(\|u_{1n}\|-\|u_1\|)
\leq \langle \psi'(u_{1n},u_{2n})-\psi'(u_1,u_2),(u_1-u_{1n},0)\rangle,
\]
which together with \eqref{14} yields  
$\|u_{1n}\|\to \|u_1\|$. It follows that $u_{1n}\to u_1$ strongly in
$W^{1,p}_{T}$ by the uniform convexity of $W^{1,p}_{T}$. Similarly,
we have $u_{2n}\to u_2$ strongly in $W^{1,q}_{T}$. Hence,
the $(PS)$ condition is satisfied.

Let $\widetilde{W} = \widetilde{W}_T^{1,q} \times
\widetilde{W}_T^{1,p}$ be the subspace of $W$ given by
\[
\widetilde{W}= \{ (u_1,u_2) \in W ~ \mid ~ (\bar{u}_1, \bar{u}_2)
= (0,0) \}.
\]
Then
\begin{equation}\label{6}
\varphi (u_1, u_2) \to +\infty
\end{equation}
as $\|(u_1,u_2)\|_W \to \infty$ in $\widetilde{W}$. In fact, by
the proof of Theorem \ref{Th1}, one has
\begin{align*}
\varphi(u_1,u_2)
&=  \frac{1}{q} \int_0^T |\dot{u}_1(t)|^q dt +
\frac{1}{p} \int_0^T |\dot{u}_2(t)|^p dt\\
&\quad + \int_0^T (F(t,u_1(t),u_2(t)) - F(t, \bar{u}_1,\bar{u}_2)
) dt + \int_0^T F(t, \bar{u}_1,\bar{u}_2) dt \\
&\geq \frac{1}{2q}\|\dot{u}_1\|_q^q+\frac{1}{4p}
\|\dot{u}_2\|_p^p - 2^{q-1}C_1^{\alpha_1 +1}\|f_1\|_{L^1}
 \|\dot{u}_1\|_q^{\alpha_1 +1}\\
&\quad- 2^{p-1}C_1^{\alpha_2 +1}\|f_2\|_{L^1}\|\dot{u}_2\|_p^{\alpha_2 +1}
 -\frac{4^{ p'/p}(2^{\beta_2}\|g_2\|_{L^1}C_1^{\beta_2+1})^{p'}
   }{p'} \|\dot{u}_1\|_q^{\beta_2p'}\\
&\quad  - C_1\|h_1\|_{L^1}\|\dot{u}_1\|_q-   C_1\|h_2\|_{L^1}\|\dot{u}_2\|_p 
+\int_0^T F(t, \bar{u}_1,\bar{u}_2) dt
\end{align*}
for all $(u_1,u_2)\in \widetilde{W}$. By Wirtinger's inequality,
the norm
\[
\||(u_1,u_2) \|| = \|(\dot{u}_1, \dot{u}_2)\|_{L^q \times L^p} =
\|\dot{u}_1\|_q + \|\dot{u}_2\|_p
\]
is an equivalent norm on $\widetilde{W}$. Hence, \eqref{6} follows
from   the above inequality.

On the other hand, one has
\begin{equation}\label{7}
\varphi (x_1, x_2) \to -\infty
\end{equation}
as $|(x_1,x_2)| \to \infty$ in $ \mathbb{R}^N \times  \mathbb{R}^N$,
which follows from (H3). Now, Theorem \ref{Th2} is proved by
\eqref{6}, \eqref{7} and the Saddle Point Theorem 
(see \cite[Theorem 4.6]{rab}).
\end{proof}

\subsection*{Acknowledgements}
This research was supported by the
National Natural Science Foundation of China (No. 11071198) and the
Fundamental Research Funds for the Central Universities (No.
XDJK2010C055).

\begin{thebibliography}{00}

\bibitem{Berger-Schechter} M. S. Berger, M. Schechter; 
\emph{On the solvability of semilinear gradient operator equations}, 
Adv. Math. 25 (1977), 97-132.

\bibitem{LongYM}Y.-M. Long; 
\emph{Nonlinear oscillations for classical Hamiltonian
                 systems with bi-even subquadratic potentials}, 
Nonlinear Anal. 24 (12) (1995), 1665-1671.

\bibitem{Mawhin} J. Mawhin; 
\emph{Semi-coercive monotone variational problems}, Acad. Roy. Belg. Bull. 
Cl. Sci. 73 (1987), 118-130.

\bibitem{mw} J. Mawhin, M. Willem; 
\emph{Critical Point Theory and Hamiltonian Systems},
Springer-Verlag, Berlin/New York, 1989.

\bibitem{Pasca-PanAmer}  D. Pa\c{s}ca;  
\emph{Periodic solutions for second order differential inclusions with sublinear 
nonlinearity}, PanAmer. Math. J. 10(4) (2000), 35-45.

\bibitem{Pasca-Commun}  D. Pa\c{s}ca;  
\emph{Periodic solutions for nonautonomous second order differential inclusions 
systems with p-Laplacian}, Commun. Appl. Nonlinear Anal. 16(2) (2009), 13-23.

\bibitem{Pasca-Simon} D. Pa\c{s}ca;
\emph{ Periodic solutions of a class of nonautonomous second order diffrential 
systems with (q,p)-Laplacian}, Bull. Belg. Math. Soc. Simon
Stevin 17 (2010), no. 5, 841-850.

\bibitem{Pasca-Singap}  D. Pa\c{s}ca; 
\emph{Periodic solutions of second-order diffrential inclusions systems with 
$(q,p)$-Laplacian}, Anal. Appl. (Singap.) 9 (2011), no. 2, 201-223.

\bibitem{P-Tang} D. Pa\c{s}ca, C.-L. Tang; 
\emph{Some existence results on periodic solutions of nonautonomous second-order 
differential systems with ($q,p$)-Laplacian}, Appl. Math. Lett. 23 (2010), 246-251.

\bibitem{Pasca-Tang} D. Pa\c{s}ca, C.-L. Tang; 
\emph{Some existence results on periodic solutions of ordinary $(q,p)$-Laplacian 
systems}, J. Appl. Math. Inform. 29 (2011), no. 1-2, 39-48.

\bibitem{rab} P. H. Rabinowitz;
 \emph{Minimax methods in critical point theory with
applications to differential equations}, CBMS Reg. Conf. Ser. in
Math. No. 65, AMS, Providence, RI, 1986.

\bibitem{Tang1} C.-L. Tang; 
\emph{Periodic solutions of nonautonomous second order systems with 
$\gamma$-quasi\-subadditive potential}, J. Math. Anal. Appl. 189 (1995), 671-675.

\bibitem{Tang2} C.-L. Tang; 
\emph{Periodic solutions of nonautonomous second order systems}, 
J. Math. Anal. Appl. 202 (1996), 465-469.

\bibitem{tang} C.-L. Tang; 
\emph{Periodic solutions for nonautonomous second order
systems with sublinear nonlinearity}, Proc. Amer. Math. Soc. 126(11) (1998), 
3263-3270.

\bibitem{Tang-Wu} C.-L. Tang, X. P. Wu; 
\emph{Periodic solutions for second order systems with not uniformly coercive 
potential}, J. Math. Anal. Appl. 259 (2001), 386-397.

\bibitem{Tang-Meng} X. H. Tang, Q. Meng; 
\emph{Solutions of a second-order Hamiltonian system with periodic boundary 
conditions}, Nonlinear Anal. 11 (2010) 3722-3733.

\bibitem{tg} Y. Tian, W. Ge; 
\emph{Periodic solutions of non-autonomous second-order
systems with a $p$-Laplacian}, Nonlinear Anal. 66 (1) (2007),
192-203.

\bibitem{Willem}  M. Willem; 
\emph{Oscillations forc\'{e}es de syst\`{e}mes hamiltoniens}, in: 
Public. S\'{e}min. Analyse Non Lin\'{e}aire, Univ. Besancon, 1981.

\bibitem{Wu-Tang} X.-P. Wu, C.-L. Tang; 
\emph{Periodic solutions of a class of nonautonomous second order systems}, 
J. Math. Anal. Appl. 236 (1999), 227-235.

\end{thebibliography}

\end{document}