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

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2006(2006), No. 134, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2006/134\hfil Periodic solutions]
{Periodic solutions for second-order Hamiltonian systems with the
p-Laplacian}

\author[Y. Tian, W. Ge\hfil EJDE-2006/134\hfilneg]
{Yu Tian, Weigao Ge} 

\address{Yu Tian \newline
 School of Science, Beijing University of Posts and
Telecommunications, Beijing 100876, China}
\email{tianyu2992@163.com}

\address{Weigao Ge \newline
 Department of Applied Mathematics, 
 Beijing Institute of Technology, Beijing 100081, China}
\email{gew@bit.edu.cn}

\date{}
\thanks{Submitted July 16, 2006. Published October 19, 2006.}
\thanks{Supported by grants 10671012 from the National Natural
Sciences Foundation of China and  \hfill\break\indent
 20050007011 from
Foundation for Ph.D. Specialities of Educational Department of China.}
\subjclass[2000]{34B15, 58E30} 
\keywords{ Hamiltonian system; p-Laplacian; Mountain Pass Theorem; \hfill\break\indent 
Periodic solution}


\begin{abstract}
 In this paper, we  investigate the periodic solutions of
 Hamiltonian system with the p-Laplacian. By using Mountain Pass
 Theorem the existence of at least  one periodic solution is obtained,
 Furthermore, under suitable assumptions, we obtain the existence of
 infinitely many solutions via $Z_2$-symmetric  version of the
 Mountain Pass Theorem.
\end{abstract}

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


\section{Introduction}

There has been published  an extensive literature related to the
existence of the periodic solutions of second-order differential
equations (systems) recently; see for example the refeences  in this
article and the references cited therein. In \cite{Bahri,benci} the
authors considered the system
 \begin{equation}
\begin{gathered}
 \ddot{u}(t)+\nabla H(t, u(t))=0\quad \text{a.e. } t\in[0, T],\\
 u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{gathered}\label{01}
\end{equation}
 and  obtained multiple solutions
under the following assumption on the potential $H$: There exist
$R>0$, $\theta\in]0, 1/2[$ such that
\begin{equation}
0<H(t, u)\le \theta \nabla _{u}H(t, u)u\label{1}
\end{equation}
for each $t\in [0, T]$, for each $u\in R^k, |u|\ge R$. When $H(t,
u(t))=b(t)V(u(t))$, $b\in C([0, T], R)$ and $b$ changes its sign,
 there are many
existence results of nontrivial periodic solutions for problem
\eqref{01} (see  \cite{Antonacci,btpositive1,btpositive2}).
 All of them assumed that
\[
\int_{0}^{T}b(t)dt\neq 0.
\]
 Ding \cite{Y.H.Ding} established the existence of
periodic solutions for
\begin{equation}
\int_{0}^{T}b(t)dt=0\label{111}
\end{equation}
under the so-called global Ambrosetti-Rabinowitz condition, that is,
there exists a constant $\theta\in]0, \frac{1}{2}[$ such that
\eqref{1} holds for all $t\in [0, T]$ and $ u\in
\mathbb{R}^N\setminus\{0\}$.
 In the case of \eqref{111}, Chen and Long \cite{chen}
 obtained the existence of one solution by the Saddle Point Theorem.
Very recently, Tang and Wu \cite{tangwu} generalized the results in
\cite{chen} and got the following
 theorem with the aid of generalized Mountain Pass Theorem.

\begin{theorem} \label{thmA}
 Suppose $\mu >2$, $b\in C([0, T], R)$ satisfying
$\int_{0}^{T}b(t)dt=0$, $b\not\equiv0$ and $H: [0, T] \times
\mathbb{R}^N\to R$, $H(t, x)$ is measurable in $t$ for every $x\in
\mathbb{R}^N$ and continuously differentiable in $x$
 for a.e. $t\in[0, T]$, such that
\begin{itemize}
\item[(A1)] $\int_{0}^{T}H(t, x)dt\ge 0$ for all $x\in \mathbb{R}^N$.

\item[(A2)] There exist $g\in L^{1}(0, T)$, $\alpha_0\in (0, \omega^2/2)$ and
 $r_0>0$ such  that
  $|\nabla H(t, x)|\le g(t)$
 for all $x\in \mathbb{R}^N$ and a.e. $t\in[0, T]$.

\item[(A3)] $|H(t, x)|\le \alpha_0|x|^2$
 for all $|x|\le r_0$ and a.e. $t\in[0, T]$, where $\omega=2\pi/T$.
\end{itemize}
Then the problem
\begin{gather*}
 \ddot{u}(t)+b(t)|u(t)|^{\mu-2}u(t)+\nabla H(t, u(t))=0\quad
\text{a.e. }t \in [0, T],\\
 u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{gather*}
 has at least one nonzero solution.
\end{theorem}

  However, in [1-14, 16, 17], the highest order derivatives are linear.
But so far, few papers discuss periodic solutions for  second order
system with the p-Laplacian. On the other hand, it is well known
that the study of the existence for quasilinear differential
equations is very important. Motivated by the above works, we
consider the existence of solutions for the following second-order
Hamiltonian system with
 p-Laplacian:
\begin{equation}\label{000}
\begin{gathered}
\frac{d}{dt}(\Phi_p(\dot{u}(t)))+B(t)\Phi_{\mu}(u(t))
+\nabla H(t, u(t))=0,\quad t\in[0, T],\\
u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{gathered}
\end{equation}
where $p>1$, $\Phi_p(u):=(|u_1|^{p-2}u_1, \dots, |u_N|^{p-2}u_N)$,
$u=(u_1, \dots, u_N)$, $\mu>p$, $T>0$, $H: [0,T]\times
\mathbb{R}^N\to R$, $H(t, x)$ is measurable in $t$ for every $x\in
\mathbb{R}^N$ and continuously differentiable in $x$ for a.e.
$t\in[0, T]$, $\nabla H(t, x)=(\partial H/\partial x)(t, x)$.
\[
B(t)=\begin{bmatrix}
b_1(t)\\
& &\ddots\\
& & & b_N(t)
\end{bmatrix},
\]
 $b_i\in C([0, T], R)$, $b_i\not\equiv 0$, $i=1, 2,\dots, N$.
For $p=2$, $\frac{d}{dt}(\Phi_p(\dot{u}(t)))=\ddot{u}(t)$. To apply
critical point theory to \eqref{1}, it is necessary to check that
corresponding functional verifies the Palais-Smale condition
((PS)-condition). Taking the quasi-linear
 into consideration, uniformly convex of $L^p$ and related
differential inequalities have to be used (see section 2).

 Using the Mountain Pass Theorem,  the existence of at
least one solution is obtained. Furthermore, under the hypothesis of
eveness of the functional, the existence of infinitely many
solutions is obtained by using $Z_2$ version of the Mountain Pass
Theorem.

  The following lemmas are taken from \cite{zhuanzhu}, and
 will be useful in the proofs of our main results.

\begin{lemma}\label{th3}
Let  $E$ be a real Banach space with $E=V\oplus X$, where $V$ is
finite dimensional. Suppose $I\in C^{1}(E, R)$ satisfies (PS), and
\begin{itemize}
\item[(I1)] There are constants $\rho, \alpha>0$ such that $I_{\partial
B_{\rho}\cap X}\ge \alpha$, and
\item[(I2)] There is an $e\in \partial B_1\cap X$ and $R>\rho$ such that
if $Q\equiv (\overline{B}_R\cap V)\oplus \{re:0<r<R\}$, then
$I_{\partial Q}\le 0$.
\end{itemize}
Then $I$ possesses a critical value $c\ge \alpha$ which can be
characterized as
\[
c\equiv \inf_{h\in \Gamma}\max_{u\in Q}I(h(u)),
\]
where $\Gamma=\{h\in C(\overline{Q}, E):h=id\mbox{ on } \partial
Q\}$, where $id$ denotes the identity operator.
\end{lemma}

\begin{lemma}\label{th3.2}
Let $E$ be an infinite dimensional real Banach space and let $I\in
C^1(X, R)$ be even, satisfying the (PS) condition and $I(0)=0$. If
$E=V\oplus X$, where $V$ is finite dimensional, and $I$ satisfies
 \begin{itemize}
\item[(J1)] there exist constants $\rho, \alpha>0$ such that
$I|_{\partial B_{\rho}\cap X}\ge \alpha$ and
\item[(J2)] for each finite dimensional subspace $V_1\subset E$, the set
$\{x\in V_1: I(x)\ge 0\}$ is bounded.
\end{itemize}
Then $I$ has an unbounded sequence of critical values.
\end{lemma}

\section{Preliminaries}

 In the proof of main results, we will need the following
preliminary results.
 For convenience, let
\[
W_{T}^{1, p}([0, T])=\big\{u: [0, T]\mapsto \mathbb{R}^N: u \mbox{
is abs. cont., } u(0)=u(T),\; \dot{u}\in L^p(0, T;
\mathbb{R}^N)\big\}
\]
 be a Sobolev space with the norm
\[
\|u\|_{W_{T}^{1,
p}}=\Big(\int_{0}^{T}|u(t)|^p+|\dot{u}(t)|^pdt\Big)^{1/p}.
\]
  For any $x\in W_{T}^{1, p}([0,T])$, define $A: W_{T}^{1, p}([0, T])\to R$ by
\begin{equation}\label{A}
A(x)=\int_{0}^{T}\sum_{i=1}^{N}|\dot{x}_{i}(t)|^{p}dt
=\sum_{i=1}^{N}\|\dot{x}_{i}\|_{L^p}^{p}.
\end{equation}
Let $A_i(x)=\|\dot{x}_{i}\|_{L^p}^{p}$, so
$A(x)=\sum_{i=1}^{N}A_i(x)$. Clearly, $A$ is convex. Now we claim
that $A$ is lower semi-continuous on $W_{T}^{1, p}([0, T])$. So $A$
is weakly lower semi-continuous on $W_{T}^{1, p}([0, T])$ (see
\cite[Theorem 1.2]{mawhin}). In fact, we only need to show $A_i$,
$i\in\{1, 2, \dots, N\}$ is weakly lower semi-continuous on
$W_{T}^{1, p}([0, T])$. Let $x_n\to x$ in $W_{T}^{1, p}([0,T])$,
from which it follows $\|\dot{x}_n-\dot{x}\|_{L^p}\to0$. By
\[
\|x+y\|_{L^p}\le \|x\|_{L^p}+\|y\|_{L^p}\quad \forall x, y\in
L^{p}([0, T]),
\]
we have
\[
A_i(x_n)-A_i(x)=\|\dot{x}_{ni}\|_{L^p}^p-\|\dot{x}_{i}\|_{L^p}^{p}\le
\left(\|\dot{x}_{ni}-\dot{x}_i\|_{L^p}+\|\dot{x}_i\|_{L^p}\right)^{p}
-\|\dot{x}_i\|_{L^p}^{p}\to 0
\]
 as $n\to\infty$, and
\begin{align*}
A_i(x_n)-A_i(x)
&=\|\dot{x}_{ni}\|_{L^p}^p-\|\dot{x}_{i}\|_{L^p}^{p}\\
&\ge \Big(\Big|\|\frac{\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p}
-\|\frac{\dot{x}_{ni}-\dot{x}_i}{2}\|_{L^p}\Big|\Big)^p\\
& \quad - \left(\|\frac{\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p}
-\|\frac{-\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p}\right)^p\to 0\quad
\mbox{as } n\to\infty.
\end{align*}
 Therefore,
\[
\lim_{n\to\infty}A_i(x_n)=A_i(x).
\]

\begin{lemma}[\cite{uniform}] \label{le1}
For the space $L^{p}([0, T])$, the following inequalities between
the norms of two arbitrary elements $x$ and $y$ of the space are
valid (here $q$ is the conjugate index, $q=p/(p-1)$):
\begin{gather}\label{ineq1}
\|\frac{x+y}{2}\|_{L^p}^{p}+\|\frac{x-y}{2}\|_{L^p}^{p}\le
\frac{1}{2}(\|x\|_{L^{p}}^{p}+\|y\|_{L^p}^{p})\quad
\mbox{for }p\ge2,\\
\label{ineq2}
\|\frac{x+y}{2}\|_{L^p}^{q}+\|\frac{x-y}{2}\|_{L^p}^{q}\le
\big[\frac{1}{2}(\|x\|_{L^{p}}^{p}+\|y\|_{L^p}^{p})\big]^{q-1}\quad
\mbox{for }1<p<2.
\end{gather}
\end{lemma}

\begin{proposition}[\cite{napoli p-laplacian}] \label{prop1}
 Let $A$ be defined as in  \eqref{A}. Suppose that
$(x_n)_{n\in N}$  is a sequence in $W_{T}^{1, p}([0, T])$ satisfying
$x_n\rightharpoonup x$ and  the following inequality
\begin{equation}\label{ineq}
\limsup_{n\to\infty} \langle DA(x_n), x_n-x\rangle\le 0.
\end{equation}
Then $x_n\to x$ strongly in $W_{T}^{1, p}([0,T])$.
\end{proposition}

\begin{proof}
Let $x_n\rightharpoonup x$ in $W_{T}^{1, p}([0, T])$ and satisfy
\eqref{ineq}. So $(x_n)$ is bounded: $\|x\|_{W_{T}^{1, p}}\le M$,
clearly, $A_i(x_n)$ is bounded. For a subsequence, $A_i(x_n)\to
c_i$. So $A(x_n)\to c=\sum_{i=1}^{N}c_i$. By weakly lower
semi-continuous of $A$:
\[
A(x)\le \liminf_{n\to\infty}A(x_n)=c.
\]
On the other hand, since $A$ is convex and lower semi-continuous,
its graphic lies over the tangent hyper-plane at $x_n$,
\[
A(x)\ge A(x_n)+\langle DA(x_n), x-x_n\rangle.
\]
 From \eqref{ineq}, we deduce that $A(x)\ge c$, then
$A(x)=c$.

 Also we have $\frac{x+x_n}{2}\rightharpoonup x$, and again
by weakly lower semi-continuous of $A$, $A_i$,
\begin{equation} \label{**}
c=A(x)\le \liminf_{n\to\infty}A\left(\frac{x+x_n}{2}\right),\quad
c_i=A(x_i)\le\liminf_{n\to\infty}A_i\left(\frac{x+x_n}{2}\right).
\end{equation}
If we suppose that $(x_n)$ does not converge strongly to  $x$, then
there exist $\varepsilon>0$ and subsequence $(x_{nk})$ that satisfy
$\|x_{nk}-x\|_{W_{T}^{1, p}}\ge \varepsilon$. Since $(x_{n_k})$
converges uniformly to $x$ in $C([0, T])$,
$\|\dot{x}_{nk}-\dot{x}\|_{L^p}\ge \varepsilon$. So there exists
$j\in\{1,2,\dots, N\}$ and $\widetilde{\varepsilon}>0$,
$\|\dot{x}_{nkj}-\dot{x}_j\|_{L^p}\ge \widetilde{\varepsilon}$.
 From \eqref{ineq1},
\begin{align*}
\limsup_{n_k\to\infty}A\big(\frac{x+x_{nk}}{2}\big) &\le
\limsup_{n_k\to\infty}
\frac{1}{2}A(x)+\frac{1}{2}A(x_{nk})-A\big(\frac{x-x_{nk}}{2}\big) \\
&\le c- \limsup_{n_k\to\infty}\sum_{i=1}^{N}\|\frac{\dot{x}_{nki}
 -\dot{x}_i}{2}\|_{L^p}^{p} \\
&\le c-\frac{1}{2^p}\widetilde{\varepsilon}^{p},
\end{align*}
  which contradicts
\eqref{**}. For $1<p<2$, by \eqref{ineq2} we have
\begin{equation}\label{2.6}
\begin{aligned}
\limsup_{n_k\to\infty}\|\frac{\dot{x_j}+\dot{x}_{nkj}}{2}\|_{L^p}^{q}
&\le\limsup_{n_k\to\infty}
\Big[\frac{1}{2}\|\dot{x}_j\|_{L^p}^{p}+\frac{1}{2}\|\dot{x}_{nkj}
\|_{L^p}^{p}\Big]^{q-1}-\|\frac{\dot{x_j}-\dot{x}_{nkj}}{2}\|_{L^p}^{q}
\\
&\le c_{j}^{q-1}-\frac{\widetilde{\varepsilon}^{q}}{2^{q}}.
\end{aligned}
\end{equation}
By  \eqref{**} and \eqref{2.6}, we have
\[
\frac{\widetilde{\varepsilon}^{q}}{2^{q}}+c_{j}^{\frac{q}{p}}\le
c_{j}^{q-1},
\]
that is
\[
\frac{\widetilde{\varepsilon}^{pq}}{2^{pq}}+c_{j}^{q} \le
\Big(\frac{\widetilde{\varepsilon}^{q}}{2^{q}}+c_{j}^{\frac{q}{p}}\Big)^{p}
\le c_{j}^{q},
\]
a contraction. The proof is complete.
\end{proof}

\section{Main results}

  It is well known that $u$ is a $T$-periodic solution
of system \eqref{000} if and only if $u$ is a critical point in
$W_{T}^{1, p}([0, T])$ of functional $\varphi$, where
\begin{equation}\label{3.0}
\varphi(u)=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pdt
-\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}
b_i(t)|u_i(t)|^{\mu}dt-\int_{0}^{T}H(t, u(t))dt
\end{equation}
for $u\in W_{T}^{1, p}([0, T])$. $\varphi: W_{T}^{1, p}\to R$ is
well defined and $C^1$. Its derivative is given by
\begin{equation}\label{04}
\begin{aligned}
\langle\varphi'(u), v\rangle
&=\int_{0}^{T}\sum_{i=1}^{N}\Phi_p(\dot{u}_i(t))\dot{v}_i(t)dt
-\int_{0}^{T}\sum_{i=1}^{N}b_i(t)\Phi_{\mu}(u_i(t))v_i(t)dt\\
&\quad -\int_{0}^{T}(\nabla H(t, u(t)), v(t))dt,
\end{aligned}
\end{equation}
where $(\cdot, \cdot)$ is the usual inner product of $\mathbb{R}^N$.
It follows from Sobolev's inequality that
\begin{equation}
\|u\|_{\infty}\le C\|u\|_{W_{T}^{1, p}}\label{*}
\end{equation}
for all $u\in W_{T}^{1, p}([0, T])$, where
$\|u\|_{\infty}=\max_{0\le t\le T}|u(t)|$.

 \begin{theorem}\label{th1}
 Suppose $p\ge2, b_i\in C([0, T], R)$ and
$\int_{0}^{T}b_i(t)dt=0$, $b_i\not\equiv 0$, $i=1, 2, \dots, N$.
 $\int_{0}^{T}H(t, x)dt\ge0$
 for all $x\in \mathbb{R}^N$. Assume that there exist
\[
g, h\in L^1([0, T]),\quad
 \|h\|_{L^1}\le [p T^{\frac{p}{q}}\max\{N^{\frac{p}{2}-1}, 1\}]^{-1},\quad
\theta\in [0, p-1), \quad r>0
\]
such that
 \begin{equation}\label{02}|
\nabla H(t, x)|\le g(t)|x|^{\theta}
\end{equation}
 for all $x\in \mathbb{R}^N$ and a.e. $t\in[0, T]$, and
\begin{equation}\label{1.5'}
|H(t, x)|\le h(t)|x|^{p}
\end{equation}
for $|x|\le r$ and a.e. $t\in[0, T]$. Then system \eqref{000} has at
least one nonzero solution.
\end{theorem}

\begin{proof} The proof is divided into three steps.

\noindent {\it Step 1.} We claim that the functional $\varphi$
satisfies the Palais-Smale condition, that is, $(u_n)$ has a
convergent subsequence whenever it satisfies $\varphi'(u_n)\to 0$ as
$n\to \infty$ and $\{\varphi(u_n)\}$ is bounded.

 First we prove that $(u_n)$ is a bounded sequence in
$W_{T}^{1, p}([0, T])$. Suppose that $(u_n)$ is unbounded. Passing
to a subsequence, we may assume if necessary, that
$\|u_n\|_{W_{T}^{1, p}}\to\infty$ as $n\to\infty$. Set
$v_n=\frac{u_n}{\|u_n\|_{W_{T}^{1, p}}}$. Then $(v_n)$ is bounded so
that it has a subsequence, say $(v_n)$, which weakly converges to
$v_0$. By \cite[Proposition 1.2]{mawhin}, $(v_n)$ converges to $v_0$
uniformly on $[0, T]$. Hence one has
$\overline{v}_n\to\overline{v}_0$, where
$\overline{v}=\frac{1}{T}\int_{0}^{T}v(s)ds$.
      It follows from \eqref{*}  and \eqref{02} that
\begin{equation}\label{*)}
\begin{aligned}
&\mu\int_{0}^{T}H(t, u_n(t))dt-\int_{0}^{T}(\nabla H(t, u_n(t)), u_n(t))dt\\
&\le \mu\int_{0}^{T}\left(\int_{0}^{1} \left(\nabla H(t, su_n(t)),
u_n(t)\right)ds\right)dt
 +\mu\int_{0}^{T}H(t, 0)dt \\
&\quad  +\int_{0}^{T}|\nabla H(t, u_n(t))||u_n(t)|dt\\
&\le\frac{\mu}{\theta+1}\int_{0}^{T}g(t)|u_n(t)|^{\theta+1}dt
  +\mu\int_{0}^{T}H(t,0)dt+\int_{0}^{T}g(t)|u_n(t)|^{\theta+1}dt\\
&\le
\big(\frac{\mu}{\theta+1}+1\big)\|g\|_{L^1}C^{\theta+1}\|u_n\|_{W_{T}^{1,
p}}^{\theta+1}+\mu\int_{0}^{T}H(t,0)dt.
\end{aligned}
\end{equation}
 Then by \eqref{3.0}, \eqref{04}, \eqref{*)}, we have
\begin{align*}
&\big(\frac{\mu}{p}-1\big)\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_{ni}(t)|^{p}dt\\
&=\mu \varphi(u_n)-\langle\varphi'(u_n), u_n\rangle +\int_{0}^{T}\mu
H(t, u_n)dt
-\int_{0}^{T}(\nabla H(t, u_n), u_n)dt\\
&\le \mu \varphi(u_n)-\langle\varphi'(u_n),
u_n\rangle+\big(\frac{\mu}{\theta+1}+1\big)\|g\|_{L^1}C^{\theta+1}
\|u_n\|_{W_{T}^{1, p}}^{\theta+1}+\mu\int_{0}^{T}H(t, 0)dt,
\end{align*}
which implies $\|\dot{v}_{ni}\|_{L^p}\to 0$ as $n\to\infty$, $i=1,
2, \dots, N$. So $\widetilde{v}_n\to 0$ in $W_{T}^{1, p}([0, T])$ as
$n\to\infty$, thus $v_n\to \overline{v}_0$ as $n\to\infty$, where
$\widetilde{v}=v-\overline{v}$. Hence $v_0=\overline{v}_0$ and
$\|v_0\|_{W_{T}^{1, p}}=1$. On the other hand, from \eqref{04},
\begin{align*}
&\Big|\int_{0}^{T}(B(t)\Phi_{\mu}(v_n(t)), v(t))dt\Big| \\
&=\Big|\int_{0}^{T}\sum_{i=1}^{N}b_i(t)|v_{ni}(t)|^{\mu-2}v_{ni}(t)v_i(t)dt
\Big|\\
&\le \|u_n\|_{W_{T}^{1, p}}^{1-\mu}|\langle\varphi'(u_n),
v\rangle|+\|u_n\|_{W_{T}^{1,
p}}^{p-\mu}\Big|\int_{0}^{T}\sum_{i=1}^{N}|\dot{v}_{ni}(t)|^{p-2}
\dot{v}_{ni}(t)\dot{v}_i(t)dt\Big| \\
&\quad +\|u_n\|_{W_{T}^{1, p}}^{1-\mu}
\Big|\int_{0}^{T}(\nabla H(t, u_n(t)),v(t))dt\Big|\\
&\le \|u_n\|_{W_{T}^{1,p}}^{1-\mu}\|\varphi'(u_n)\|\|v\|
 +\|u_n\|_{W_{T}^{1,p}}^{p-\mu}\sum_{i=1}^{N}\|\dot{v}_{ni}\|_{L^p}^{p-1}
 \|\dot{v}_i\|_{L^p}\\
&\quad +C^{\theta}\|u_n\|_{W_{T}^{1,p}}^{1-\mu+\theta}
 \|g\|_{L^1}\|v\|_{\infty},
\end{align*}
as $n\to\infty$, which implies that
\[
\Big|\int_{0}^{T}(B(t)\Phi_{\mu}(v_0(t)), v(t))dt\Big|=0
\]
for all $v\in W_{T}^{1, p}([0, T])$. By the arbitrariness of $v$,
one has
\[
B(t)\Phi_{\mu}(v_0(t))=0
\]
for a.e. $t\in[0, T]$. Because $v_0=\overline{v}_0\neq 0$, we have
$b_i(t)=0$ for a.e. $t\in[0, T]$. It follows from the continuity of
$b_i$ that $b_i=0$ for $t\in[0, T]$, which contradicts the condition
$b_i\not\equiv 0$. Hence $(u_n)$ is a bounded sequence.

 From the reflexivity of $W_{T}^{1, p}([0, T])$, we
extract a weakly convergent subsequence, that for simplicity, we
call $(u_n), u_n\rightharpoonup u$. Following we will show that
$(u_n)$ converges strongly to $u$. To this end, we note that
$A(u)=\sum_{i=1}^{N}\int_{0}^{T}|\dot{u}_i(t)|^{p}dt$. From
Proposition \ref{prop1} it is enough to prove
\[
\limsup_{n\to\infty}\int_{0}^{T}(DA(u_n), u_n-u)dt
=\limsup_{n\to\infty}\int_{0}^{T}\sum_{i=1}^{N}
\Phi_p(\dot{u}_{ni}(t))(\dot{u}_{ni}(t)-\dot{u}_{i}(t))dt\le 0.
\]
 From \eqref{04}, it follows
\begin{align*}
&\int_{0}^{T}\sum_{i=1}^{N}\Phi_p(\dot{u}_{ni}(t))(\dot{u}_{ni}(t)
 -\dot{u}_i(t))dt\\
&= \langle\varphi'(u_n), u_n-u\rangle +\int_{0}^{T}
  \sum_{i=1}^{N}b_i(t)\Phi_{\mu}(u_{ni}(t))(u_{ni}(t)-u_i(t))dt\\
&\quad +\int_{0}^{T}(\nabla H(t, u_n(t)), u_n(t)-u(t))dt.
\end{align*}
Now since the Sobolev embedding $W_{T}^{1, p}([0, T])\hookrightarrow
C([0, T])$ is compact, we get (for a subsequence) that $u_n\to u$
uniformly in $C([0, T])$. Since $\varphi'(u_n)\to 0$ by assumption,
and $u_n-u$ is bounded in $W_{T}^{1, p}([0, T])$, we deduce that
$\langle\varphi'(u_n), u_n-u\rangle\to 0$. Moreover,
\begin{align*}
&\int_{0}^{T}\sum_{i=1}^{N} b_i(t)\Phi_{\mu}(u_{ni}(t))(u_{ni}(t)-u_i(t))dt\\
&\le T\sum_{i=1}^{N}\max_{t\in[0, T]}
b_i(t)\Phi_{\mu}(\|u_{ni}\|_{\infty})\|u_{ni}-u_i\|_{\infty} \to 0
\end{align*}
and
\begin{align*}
& \int_{0}^{T}(\nabla H(t, u_n), u_n-u)dt\\
&\le  \int_{0}^{T}g(t)|u_n|^{\theta}(u_n-u)dt\le\|g\|_{L^1}
\|u_n\|_{\infty}^{\theta}\|u_n-u\|_{\infty}\to 0.
\end{align*}
So
$\limsup_{n\to\infty}\int_{0}^{T}\sum_{i=1}^{N}\Phi_p(\dot{u}_{ni})
(\dot{u}_{ni}-\dot{u}_i)dt=0$ and from Proposition \ref{prop1},
$u_n\to u$ strongly in $W_{T}^{1, p}([0, T])$.

\noindent{\it Step 2.} Let $W_{T}^{1, p}=\mathbb{R}^N\oplus
\widetilde{W}_{T}^{1, p}$, where
\[
\widetilde{W}_{T}^{1, p}([0, T])=\{u\in \widetilde{W}_{T}^{1, p}([0,
T]): \int_{0}^{T}u(t)dt=0\}.
\]
We claim there exist $\rho>0, \alpha>0$ such that $\varphi(u)\ge
\alpha$ for all $u\in S=\{u\in \widetilde{W}_{T}^{1, p}([0, T]):
\|u\|_{W_{T}^{1, p}}=\rho\}$.

 By \eqref{1.5'},  for $u\in \widetilde{W}_{T}^{1,p}([0, T])$, we have
\begin{equation}\begin{aligned}
\int_{0}^{T}H(t, u)dt &\le
\int_{0}^{T}|h(t)|dt\|u\|_{\infty}^{p}=\|h\|_{L^1}
\Big(\sum_{i=1}^{N}\|u_i\|_{\infty}^{2}\Big)^{p/2}\\
&\le \|h\|_{L^1}\Big[\sum_{i=1}^{N}
\Big(\int_{0}^{T}|\dot{u}_i(t)|dt\Big)^2\Big]^{p/2} \\
&\le \|h\|_{L^1}\max\{N^{\frac{p}{2}-1},
1\}\sum_{i=1}^{N}\left(\int_{0}^{T}|\dot{u}_i(t)|dt\right)^p\\
&\le T^{\frac{p}{q}} \|h\|_{L^1}\max\{N^{\frac{p}{2}-1},
1\}\sum_{i=1}^{N}\|\dot{u}_i\|_{L^p}^{p}.
\end{aligned}\label{''}
\end{equation}
 From  \eqref{1.5'} and \eqref{''} it follows that
\begin{align*}
\varphi(u)&=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^{p}dt-\frac{1}{\mu}\int_{0}^{T}
\sum_{i=1}^{N}b_i(t)|u_i(t)|^{\mu}dt-\int_{0}^{T}H(t,
u(t))dt\\
&\ge
\Big(\frac{1}{p}-\|h\|_{L^1}T^{\frac{p}{q}}\max\{N^{\frac{p}{2}-1},
1\}\Big)\sum_{i=1}^{N}\|\dot{u}_i\|_{L^p}^{p}-\frac{T^{1+\mu/q}}{\mu}
\sum_{i=1}^{N}\|b_i\|_{\infty} \|\dot{u}_i\|_{L^p}^{\mu}
\end{align*}
for all $u\in \widetilde{W}_{T}^{1, p}([0, T])$. Since $\mu>p$, we
can choose $\rho>0$ small enough such that $\varphi(u)\ge \alpha>0$.

\noindent {\it Step 3.} By $\int_{0}^{T}b_i(t)dt=0$, there exist
$t_{i1}, t_{i2}\in[0, T]$ such that $b_i(t)\ge 0$ for $t\in[t_{i1},
t_{i2}]$. Choose $e_i\in \widetilde{W}_{T}^{1,p}([0, T])$ such that
$e_i(t)\equiv 0$ for all $t\in[0,t_{i1}]\cup[t_{i2}, T]$,
$e_i\not\equiv 0$ and $\int_{0}^{T}b_i(t)e_i(t)dt=0, i=1, 2,\dots,
N$. Now $e=(e_1, e_2, \dots, e_N)\in \widetilde{W}_{T}^{1, p}([0,
T])$. Since $\mathbb{R}^N$ is definite dimensional, we will show
that there exists $R>\rho>0$ such that
\[
\varphi|_{\partial Q}\le 0,\quad Q\equiv\{re:r\in[0, R]\} \oplus
(B_{R}\cap \mathbb{R}^N).
\]
 To this end, we denote
\begin{gather*}
\varphi_1(u)=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^{p}dt,\quad
\varphi_2(u)=-\int_{0}^{T}\sum_{i=1}^{N}b_i(t)|u_i(t)|^{\mu}dt,\\
\varphi_3(u)=-\int_{0}^{T}H(t, u)dt.
\end{gather*}
Then $\varphi=\varphi_1+\varphi_2+\varphi_3$. By $\sigma\in
\mathbb{R}^N$, $\int_{0}^{T}b_i(t)dt=0$ and
$\int_{0}^{T}b_i(t)e_i(t)dt=0$ we have
\begin{equation}\begin{aligned}
\int_{t_{i1}}^{t_{i2}}b_i(t)(|\sigma_i|^2+|re_i|^2)dt&=
\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i|^2dt\\
&\le\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i|^{\mu}dt\Big)^{2/\mu}
\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{\mu-2}{\mu}}.
\end{aligned}\label{3.3}
\end{equation}
Then one has
\begin{equation}
\begin{aligned}
&\varphi_2(\sigma+re)\\
&=-\sum_{i=1}^{N}\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i(t)|^{\mu}dt
 +\sum_{i=1}^{N}\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i|^{\mu}dt\\
&\le-\sum_{i=1}^{N}\Big[\int_{t_{i1}}^{t_{i2}}b_i(t)(|\sigma_i|^2
 +r^2|e_{i}(t)|^2)dt
\Big]^{\mu/2}\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{2}}\\
&\quad +\sum_{i=1}^{N}|\sigma_i|^{\mu}
\int_{t_{i1}}^{t_{i2}}b_i(t)dt\\
&=-\sum_{i=1}^{N}\Big[\left(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\right)^{2/\mu}
|\sigma_i|^2+r^2
\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^{2}dt\left(\int_{t_{i1}}^{t_{i2}}
b_i(t)dt\right)^{\frac{2-\mu}{\mu}}\Big]^{\mu/2}\\
&\quad +\sum_{i=1}^{N}|\sigma_i|^{\mu}\int_{t_{i1}}^{t_{i2}}b_i(t)dt
\end{aligned}\label{3.4}
\end{equation}
for $\sigma\in \mathbb{R}^N, e\in \widetilde{W}_{T}^{1, p}$. Because
$\int_{0}^{T}H(t, u)dt>0$ for all $u\in \mathbb{R}^N$, we have
\begin{equation}
\begin{aligned}
\varphi_3(\sigma+re)
&= -\int_{0}^{T}H(t, \sigma+re(t))dt\\
&\le -\min\Big\{\int_{0}^{T}H(t,q)dt: q\in \mathbb{R}^N\Big\}\\
&\le 0.
\end{aligned} \label{3.5}
\end{equation}
Therefore, by \eqref{3.4} and \eqref{3.5}, there exist $0<L, m_i,
N<\infty$ such that
\begin{equation}\begin{aligned}
&\varphi(\sigma+re)\\
&=\varphi_1(\sigma+re)+\varphi_2(\sigma+re)+\varphi_3(\varrho+re)\\
&\le \frac{r^p}{p}\sum_{i=1}^{N}\int_{0}^{T}|\dot{e}_i(t)|^pdt+
\sum_{i=1}^{N}|\sigma_i|^{\mu}\int_{t_{i1}}^{t_{i2}}b_i(t)dt\\
&\quad
-\sum_{i=1}^{N}\Big[\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{2/\mu}|
\sigma_i|^2+r^2 \int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^{2}dt
\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{\mu}}\Big]^{\mu/2}\\
&=Lr^p+\sum_{i=1}^{N}m_i|\sigma_i|^{\mu}-\sum_{i=1}^{N}\left[m_{i}^{2/\mu}|\sigma_i|^2+r^2N\right]^{\mu/2}
\end{aligned}\label{3.5'}
\end{equation}
for all $\sigma\in \mathbb{R}^N, r\ge0$, which implies that  there
exists $R_1>\rho$ large enough such that
\begin{equation}\label{aaa}
\varphi(\sigma+re)<0\quad\mbox{ for }\|\sigma\|=R, r\in[0, R],
R>R_1.
\end{equation}
On the other hand, for $\sigma\in \mathbb{R}^N, r=R$, by
\eqref{3.4},
\begin{equation}
\begin{aligned}
&-\sum_{i=1}^{N}\Big[\ \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\
\Big)^{2/\mu}|\sigma_i|^2+R^2
\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^{2}dt
\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{\mu}}\Big]^{\mu/2}\\
&+\sum_{i=1}^{N}|\sigma_i|^{\mu}\int_{t_{i1}}^{t_{i2}}b_i(t)dt\\
&\le -R^{\mu}\sum_{i=1}^{N}
\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^2dt\Big)^{\mu/2}
\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{2}}.
\end{aligned}\label{3.6}
\end{equation}
Then by \eqref{3.5'} and \eqref{3.6},
\begin{align*}
&\varphi(\sigma+Re)\\
&\le
\frac{R^p}{p}\sum_{i=1}^{N}\int_{0}^{T}|\dot{e}_i(t)|^{p}dt-R^{\mu}\sum_{i=1}^{N}\left(\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^2dt\right)^{\mu/2}\left(\int_{t_{i1}}^{t_{i2}}b_i(t)dt
\right)^{\frac{2-\mu}{2}},
\end{align*}
which implies that there exists $R_2>R_1$ such that
\begin{equation}\label{bbb}
\varphi(\sigma+Re)\le 0\quad \mbox{for all }\sigma\in
\mathbb{R}^N,\; R>R_2.
\end{equation}
Therefore, \eqref{aaa} and \eqref{bbb} give that $\varphi|_{\partial
Q}\le 0$. Hence Theorem \ref{th1} is proved by Lemma \ref{th3}.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
It is clear that our theorem generalizes the results in [1-14, 16,
17] since $p\ge2$. Even for $p=2$, Theorem \ref{th1} generalizes
Theorem \ref{thmA} since $\nabla H(t, x)$ has more freedom in
\eqref{02} than in (A2). There are functions satisfying Theorem
\ref{th1} and not satisfying the corresponding results in
\cite{Antonacci, btpositive1, chen, Y.H.Ding, btpositive2,tangwu}.
For example, let
\begin{gather*}
b_i(t)=\sin\frac{2\pi t}{T},\quad i=1, 2\dots, N,\\
H(t, u)=\begin{cases} \frac{\pi}{4T^2}\left(\sin\frac{2\pi
t}{T}\right)\left(\frac{2}{3}|u|^{\frac{3}{2}}+\sin(|u|-1)+\frac{\pi}{6T}\right),
& |u|\ge1,\\
\frac{\pi}{4T^2}\left(\sin\frac{2\pi t}{T}\right)|u|^2,& |u|<1.
\end{cases}
\end{gather*}
 A straight forward computation shows
that $H(t, u)$ satisfies our Theorem \ref{th1} and  neither
satisfies assumptions in Theorem \ref{thmA}, nor \eqref{1}, hence
$H(t, u)$ does not satisfy the corresponding results in
\cite{Bahri,benci}. Moreover, $F(t,
x)=\frac{B(t)}{\mu}|u|^{\mu}+H(t, u)$ does not satisfy the
conditions of the results in
\cite{Antonacci,btpositive1,btpositive2} because that
$\int_{0}^{T}b_i(t)dt=0$; $\nabla F(t, x)\to 0$ as $|x|\to\infty$,
it does not satisfy \cite{chen}.
\end{remark}

\begin{remark} \label{rmk3.2} \rm
For $1<p<2$, inequalities \eqref{3.5'} and \eqref{3.6} do not hold.
\end{remark}

\begin{theorem}\label{th2}
Suppose that $p>1$, $H: [0, T]\times \mathbb{R}^N \to R$ is even
with respect to the second argument and $H(t, 0)=0$. Suppose $b_i\in
C(0, T; R^+)$ and there exist $g, h\in L^1([0, T])$, with
\[
 \|h\|_{L^1}\le [p T^{\frac{p}{q}}\max\{N^{\frac{p}{2}-1}, 1\}]^{-1},\quad
\theta\in [0, p-1), \quad r>0
\]
such that \eqref{02} \eqref{1.5'} hold. Then system \eqref{000} has
infinitely many solutions.
\end{theorem}

\begin{proof}
 Since $H$ is
even in the second argument, the functional $\varphi$ is even and
satisfying $\varphi(0)=0$. By Step 1 in the proof of Theorem
\ref{th1}, $\varphi$ satisfies (PS) condition. Let $W_{T}^{1,
p}=\mathbb{R}^N\oplus \widetilde{W}_{T}^{1, p}$. From Step 2 in the
proof of Theorem \ref{th1}, there exist $\alpha, \rho>0$ such that
$\varphi(u)\ge \alpha$ if $\|u\|=\rho$, $u\in \widetilde{W}_{T}^{1,
p}([0, T])$. Now we will verify condition (J2) in Lemma \ref{th3.2}.
\begin{equation}\begin{aligned}
&\varphi(u)\\
&=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pddt
-\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t)
|u_i(t)|^{\mu}dt-\int_{0}^{T}H(t, u(t))dt\\
&\le\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pdt-\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t)
|u_i(t)|^{\mu}dt-\int_{0}^{T}
\Big(\int_{0}^{1}(\nabla H(t,su),u)ds\Big)dt\\
&\le\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pdt
 -\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t)
|u_i(t)|^{\mu}dt+\int_{0}^{T}\frac{g(t)|u|^{\theta+1}}{\theta+1}dt.
\end{aligned}
\end{equation}
 From \eqref{*}, we have
\begin{equation}\begin{aligned}
\|u\|_{\infty}^{\theta+1}
&\le \Big(\sum_{i=1}^{N}\|u_i\|_{\infty}^{2}\Big)^{\frac{\theta+1}{2}}\\
&\le C^{\theta+1}\Big(\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1,
p}}^{2}\Big)^{\frac{\theta+1}{2}}\\
&\le C^{\theta+1}\max\{N^{\frac{\theta-1}{2}},
1\}\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{\theta+1}.
\end{aligned}\label{3.9'}
\end{equation}
 For any finite dimensional subspace $V_1\subset W_{T}^{1, p}([0, T])$, the
norm $\|\cdot\|_{W_{T}^{1,p}}$ and $\|\cdot\|_{L^{\theta}}$. So
there exists $ \overline{c}>0$ such that
\begin{equation}
\|u\|_{W_{T}^{1, p}}\le
\overline{c}\Big(\int_{0}^{T}|u(t)|^{\mu}dt\Big)^{\frac{1}{\mu}}\quad
\mbox{for }u\in V_1.\label{3.12}
\end{equation}
Moreover by H\"older inequality, there exists a positive constant
$\widetilde{c}$ such that
 \begin{equation}
 \|u\|_{L^{\theta+1}}\le \widetilde{c}\|u\|_{W_{T}^{1,p}}^{\theta+1}\label{3.13}
\end{equation}
holds. Thus \eqref{3.9'} \eqref{3.12} and \eqref{3.13} give that
there exist $0<\Gamma_i<\infty$, $i=1, 2, 3$ such that
\[
\varphi(u)\le \Gamma_1\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1,
p}}^{p}-\Gamma_2\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1,
p}}^{\mu}+\Gamma_3\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{\theta+1},
\]
which implies that  $\{x\in X_1: \varphi(x)\ge 0\}$ is bounded. Then
Lemma \ref{th3.2} can be applied to the functional $\varphi$. The
proof is completed.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
It is clear that our theorem generalizes the results in [1-14, 16,
17] since $p>1$. Even for $p=2$, the conditions of Theorem \ref{th1}
are different from the conditions in  \cite{Antonacci,Bahri,benci}
since in \cite{Antonacci} there exist $\beta>2, \alpha>0$ such that
$H(u)\ge \alpha |u|^{\beta}$ for all $u\in \mathbb{R}^N$ and in
\cite{Bahri,benci} there exists $\mu>2$ such that \eqref{1} holds
for $t\in [0, T], |x|\ge R$. There are functions satisfying Theorem
\ref{th2} but not satisfying  \cite{Antonacci, Bahri, benci}. For
example, let
\begin{gather*}
b_i(t)=t, i=1, 2, \dots, N,\quad \mu=3,\\
 H(t,u)=\begin{cases}
\frac{t}{2T^3}\left(\frac{7}{4}|u|^{\frac{7}{4}}+\sin(|u|-1)+\frac{3}{7}\right),&|u|\ge1,
\\[2pt]
\frac{|u|^2 t}{2T^3},& |u|<1. \end{cases}
\end{gather*}
Then $F(t,u)=\frac{B(t)}{3}|u|^{3}+H(t, u)$ satisfies Theorem
\ref{th2}, but does not satisfies the conditions in
\cite{Antonacci,Bahri,benci}.
\end{remark}

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\end{document}