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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 17, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/17\hfil Periodic solutions]
{Periodic solutions of non-autonomous second order systems with
$p$-Laplacian}

\author[Z. Wang, J. Zhang \hfil EJDE-2009/17\hfilneg]
{Zhiyong Wang, Jihui Zhang}  % in alphabetical order

\address{Zhiyong Wang \newline
Department of Mathematics, Nanjing University of Information
Science and Technology, Nanjing 210044, Jiangsu, China}
\email{mathswzhy1979@gmail.com}

\address{Jihui Zhang \newline
Institute of Mathematics, School of Mathematics and Computer
Sciences, Nanjing Normal University, Nanjing 210097, Jiangsu, China}
\email{jihuiz@jlonline.com}

\thanks{Submitted August 26, 2008. Published January 20, 2009.}
\subjclass[2000]{34B15, 47J30, 58E05}
\keywords{Periodic solutions; Hamiltonian system; $p$-Laplacian}

\begin{abstract}
 We prove the existence of periodic solutions for
 non-autonomous second order systems with $p$-Laplacian.
 Our main tools are the minimax methods in critical point theory.
 Our results are new, even when $p=2$.
\end{abstract}

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

\section{Introduction}

In this article concerns the existence of periodic
solutions for the  problem
\begin{equation}  \label{e1.1}
\begin{gathered}
\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))=\nabla F(t,u(t))\quad
\text{a.e. }t\in[0,T],\\
u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{gathered}
\end{equation}
where $p>1$, $T>0$, $F:[0,T]\times \mathbb{R^N}\to \mathbb{R}$
satisfies the following assumption:
\begin{itemize}
\item[(A)] $F(t,x)$ is measurable in $t$ for every
$x\in \mathbb{R^N}$ and continuously differentiable in $x$ for a.e.
$t\in[0,T]$, and there exist $a\in C(\mathbb{R^+},\mathbb{R^+}),b\in
L^1(0,T;\mathbb{R^+})$, such that
\begin{equation*}
|F(t,x)|\leq a(|x|)b(t),\quad |\nabla F(t,x)|\leq a(|x|)b(t)
\end{equation*}
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$.
\end{itemize}

Considerable attention has been paid  to the periodic solutions of
problem \eqref{e1.1} for $p=2$, in recent years. The firsts to
consider this problem when $p=2$ were Berger and Schechter
\cite{Berger} in 1977, they proved the existence of solutions to
problem \eqref{e1.1} for $p=2$ under the condition that
$F(t,x)\to\pm\infty$ as $|x|\to\infty$ uniformly for a.e. $t\in
[0,T]$. Subsequently, using the variational methods, many existence
results are obtained, we refer the readers to \cite{Mawhin89,
Tang98, Tang02, Tang01, Tang03, Zhao04, Mawhin87, Ben Naoum94,
Ekeland02, Schechter06BVP, Schechter06, Wang08} and the references
therein. However, there are few papers discussing periodic solutions
for second order systems with $p$-Laplacian. In \cite{Tian07}, Tian
has established the existence results for problem \eqref{e1.1} by
the dual least action principle. Moreover, if problem \eqref{e1.1}
with nonlinear boundary conditions, the existence of periodic
solutions has also been proved in \cite{Jebelean07} by means of the
least action principle and the mountain pass lemma.

For $p=2$, under the assumptions that there exists
$h(t)\in L^1(0,T;\mathbb{R^+})$ such that
\begin{equation*}
|\nabla F(t,x)|\leq h(t)
\end{equation*}
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$, and that
\begin{equation*}
\int_0^TF(t,x)dt\to\pm\infty\quad\text{as }
|x|\to\infty,
\end{equation*}
Mawhin and Willem in \cite{Mawhin89} have shown that problem
\eqref{e1.1} admitted a periodic solution. After that, Tang in
\cite{Tang98} generalized these results to the sublinear case.
Concertely speaking, it is assumed that the nonlinearity satisfied
the following restrictions
\begin{equation}    \label{e1.2}
|\nabla F(t,x)|\leq k(t)|x|^\alpha+p(t)\quad\text{for all } x\in
\mathbb{R^N}\text{ and a.e. } t\in[0,T],
\end{equation}
\begin{equation}    \label{e1.3}
\frac{1}{|x|^{2\alpha}}\int_0^TF(t,x)dt\to\pm\infty\quad\text{as
}|x|\to\infty,
\end{equation}
here, $k(t),p(t)\in L^1(0,T;\mathbb{R^+})$ and $\alpha\in [0,1)$.
Under these conditions, periodic solutions of problem \eqref{e1.1}
with $p=2$ have been obtained. In addition, Tang in \cite{Tang02}
first introduced the local $\alpha$-coercive conditions, that is,
there exists $q(t)\in L^1(0,T;\mathbb{R^+})$ and a subset $E$ of
$[0,T]$ with meas$(E)>0$ such that
\begin{equation}    \label{e1.4}
\frac{F(t,x)}{|x|^{2\alpha}}\leq q(t)
\end{equation}
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$, and that
\begin{equation}     \label{e1.5}
\frac{F(t,x)}{|x|^{2\alpha}}\to-\infty\quad\text{as
}|x|\to\infty
\end{equation}
for a.e. $t\in E$, to deal with the generalized Josephson-type
systems.

An interesting question naturally arises: In all the results
\cite{Mawhin89, Tang98, Tang02} discussed above, the nonlinearity is
required to grow at infinity at most like $|x|^\alpha$, is it
possible to handle nonlinearity with faster increase at infinity and
get the similar results of \cite{Mawhin89, Tang98, Tang02}. In the
present paper, we will focus on this problem and give a positive
answer. Here, we emphasize that our results are new even for $p=2$.

We now state our main theorems.

\begin{theorem} \label{thm1.1}
Suppose that $F$ satisfies assumption {\rm (A)} and the following
conditions:
\begin{itemize}
\item[(H1)] There exists a bounded nonincreasing positive function
$\omega\in C((0,\infty);\mathbb{R^+})$ with the properties:
\begin{itemize}
\item[(i)]
$\liminf_{t\to\infty}\frac{\omega(t)}{\omega(t^\gamma)}>0$\quad
for some $\gamma\in(0,1)$,
\item[(ii)]
$\omega(t)\to0,\quad\omega(t)t\to\infty$\quad as
$t\to\infty$.
\end{itemize}
Moreover, there exist $f\in L^2(0,T;\mathbb{R^+})$ and
$g\in L^1(0,T;\mathbb{R^+})$ such that
\begin{equation*}
|\nabla F(t,x)|\leq f(t)[\omega(|x|)|x|]^{p-1}+g(t)
\end{equation*}
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;

\item[(H2)] There exists a bounded non-increasing positive function
$\omega\in C((0,\infty);\mathbb{R^+})$ which satisfies the
conditions {\rm(i), (ii)} and
\begin{equation*}
\frac{1}{[\omega(|x|)|x|]^p}\int_0^TF(t,x)dx\to-\infty\quad
\text{as }|x|\to\infty.
\end{equation*}
\end{itemize}
Then problem \eqref{e1.1} has at least one solution in the set
\begin{align*}
W_T^{1,p}
:=\big\{&u:[0,T]\to \mathbb{R^N},\; u\text{ is
absolutely continuous},\\
& u(0)=u(T),\dot{u}\in L^p(0,T;\mathbb{R^N})\big\}\,.
\end{align*}
This set is a Banach space with the norm
\begin{equation*}
\|u\|:=\Big(\int_0^T|u(t)|^pdt+\int_0^T|\dot{u}(t)|^pdt\Big)^{1/p}\quad
\text{for }u\in W_T^{1,p}.
\end{equation*}
\end{theorem}

\begin{remark} \label{remark1.2}\rm
Obviously, Theorem \ref{thm1.1} does not satisfy the corresponding
conditions in \cite{Tang01, Tang03, Zhao04, Mawhin87, Ben Naoum94,
Ekeland02, Berger, Schechter06BVP, Schechter06, Wang08}.
Furthermore, there are functions $F(t,x)$ that satisfy our Theorem
\ref{thm1.1} and do not satisfy the corresponding results in
\cite{Mawhin89, Tang98, Tang02} even for $p=2$. For example, let
\begin{equation*}
\nabla
F(t,x)=-\frac{|x|^{p-2}x}{[\ln(2+|x|^2)]^{p-1}}+d(t),\quad\forall
t\in[0,T],x\in \mathbb{R^N},
\end{equation*}
where $d(t)\in L^1(0,T;\mathbb{R})$. Let
$\omega(|x|)=1/\ln(2+|x|^2),\gamma=1/2$, a straightforward
computation shows that $F(t,x)$ satisfies all the conditions of our
Theorem \ref{thm1.1}. However, it is clear that $F(t,x)$ neither
satisfies \eqref{e1.2}, \eqref{e1.3} nor \eqref{e1.4}, \eqref{e1.5} for
any $\alpha\in[0,1)$ even for $p=2$.
\end{remark}

\begin{theorem}\label{thm1.3}
Suppose that $F$ satisfies assumption {\rm (A)} and {\rm (H1)}. Moreover
assume $F$ satisfies the following conditions:
\begin{itemize}
\item[(H2*)]
\begin{itemize}
\item[(1)] There exists a bounded nonincreasing and positive function
$\omega$ in \\
$C((0,\infty);\mathbb{R^+})$ and constant $C>0$ such that
\begin{itemize}
\item[(iii)]
$\limsup_{t\to\infty}\frac{\omega(t)}{\omega(t^\gamma)}<+\infty$
for some $\gamma\in(0,1)$;
\item [(iv)]
$t\leq C\omega^2(t)t^2$ as $t\to\infty$;
\end{itemize}

\item[(2)] There exists $r(t)\in L^1(0,T;\mathbb{R^+})$ such
that
\begin{equation*}
\frac{F(t,x)}{[\omega(|x|)|x|]^p}\leq r(t)\quad\text{for all }x\in
\mathbb{R^N}\text{ and a.e. }t\in[0,T];
\end{equation*}

\item[(3)] There exists a subset $E$ of $[0,T]$ with
meas{\rm($E)>0$} such that for a.e. $t\in E$
\begin{equation*}
\frac{F(t,x)}{[\omega(|x|)|x|]^p}\to-\infty\quad\text{as }|x|\to\infty.
\end{equation*}
\end{itemize}
\end{itemize}
Then problem \eqref{e1.1} has at least one solution in $W_T^{1,p}$.
\end{theorem}

\begin{remark} \label{remark1.4}\rm
There are functions $F(t,x)$ that satisfy our Theorem
\ref{thm1.3} and not do not satisfy the corresponding results in
\cite{Mawhin89, Tang98, Tang02, Tang01, Tang03, Zhao04, Mawhin87,
Ben Naoum94, Ekeland02, Berger, Schechter06BVP, Schechter06,
Wang08}. For example, let
\begin{equation*}
F(t,x)=-\theta(t)\frac{|x|^p}{[\ln(2+|x|^2)]^{p-1}},\quad\forall
t\in[0,T],x\in \mathbb{R^N},
\end{equation*}
where
\begin{equation*}
\theta(t)=\begin{cases}
 \sin\frac{2\pi t}{T}, &t\in[0,T/2],\\
 0, &t\in[T/2,T].
 \end{cases}
\end{equation*}
Take $\omega(|x|)=1/\ln(2+|x|^2)$, $\gamma=1/2$, $E=[T/6,T/4]$, by simple
computation, $F(t,x)$ satisfies our Theorem \ref{thm1.3}.
However, here $F(t,x)$ also neither satisfies \eqref{e1.2},
\eqref{e1.3} nor \eqref{e1.4}, \eqref{e1.5} for any $\alpha\in[0,1)$
even for $p=2$.
\end{remark}

\section{Preliminaries}

For convenience, we will denote various positive constants as
$C_i$, $i=0,1,2,\dots$. Define functional $\varphi$ on $W_T^{1,p}$ by
\begin{equation}  \label{e2.1}
\varphi(u)=\frac{1}{p}\int_0^T|\dot{u}(t)|^pdt+\int_0^TF(t,u(t))dt
\end{equation}
for $u\in W_T^{1,p}$. It follows from assumption (A) that functional
$\varphi$ is continuously differentiable on $W_T^{1,p}$, moreover,
one has
\begin{equation*}
(\varphi'(u),v)=\int_0^T [(|\dot{u}(t)|^{p-2}\dot{u}(t),\dot{v}(t))
+(\nabla F(t,u(t)),v(t))]dt
\end{equation*}
for all $u,v\in W_T^{1,p}$. It is well known that the solutions to
problem \eqref{e1.1} correspond to the critical points of the
functional $\varphi$.

For $u\in W_T^{1,p}$, let $\bar{u}:=\frac{1}{T}\int_0^Tu(t)dt$ and
$\tilde{u}(t):=u(t)-\bar{u}$, then we have
\begin{align*}
& \|\tilde{u}\|_\infty\leq
C_0\|\dot{u}\|_{L^p}\quad\text{(Sobolev's inequality)},\\
& \|\tilde{u}\|_{L^p}\leq
C_0\|\dot{u}\|_{L^p}\quad\text{(Wirtinger's inequality)},
\end{align*}
where $\|\tilde{u}\|_\infty:=\max_{0\leq t\leq T}|\tilde{u}(t)|$.

To proof of our main theorems, we need the following
auxiliary results.

\begin{lemma}[\cite{Tang01}] \label{lemma2.1}
Suppose that $G$ satisfies assumption {\rm(A)} and $E$ is a
measurable subset of $[0,T]$. Assume that
\begin{equation*}
G(t,x)\to+\infty\quad\text{as }|x|\to\infty
\end{equation*}
for a.e. $t\in E$. Then for every $\delta>0$, there exists subset
$E_\delta$ of $E$ with {\rm meas(}$E\backslash E_\delta{\rm
)}<\delta$ such that
\begin{equation*}
G(t,x)\to+\infty\quad\text{as }|x|\to\infty
\end{equation*}
uniformly for all $t\in E_\delta$.
\end{lemma}

\begin{lemma}[\cite{Tang02}]  \label{lemma2.2}
For every constant $\beta>0$, there exists a constant $m_\beta>0$
such that
\begin{equation*}
\text{meas}\{t\in[0,T]||\bar{u}|<m_\beta\|\bar{u}\|\}<\beta\;\quad\text{for
all }\bar{u}\neq0.
\end{equation*}
\end{lemma}

The proof of the above lemma follows from the first part in
\cite[Lemma 3]{Tang02}, we omit the details.

\begin{lemma}  \label{lemma2.3}
Suppose that {\rm (H1)} holds, then there exists a non-increasing
positive function $\tilde{\omega}(t)\in C((0,\infty);\mathbb{R^+})$
which satisfies the following conditions:
\begin{itemize}
\item[(a)]
$\tilde{\omega}(t)\to0$, $\tilde{\omega}(t)t\to\infty$
as $t\to\infty$;

\item[(b)] $\|\nabla F(t,u(t))\|_{L^1}\leq\|f\|_{L^2}[\tilde{\omega}
(\|u\|)\|u\|]^{p-1}+\|g\|_{L^1}$;

\item[(c)] If {\rm (H2)} holds, then
\begin{equation*}
\frac{1}{[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p}\int_0^TF(t,\bar{u})dt
\to-\infty\quad \text{as }\|\bar{u}\|\to\infty;
\end{equation*}

\item[(d)] If {\rm (H2*)} holds, moreover assume that
$\limsup_{\|u\|\to\infty}\frac{\|\tilde{u}\|}{\tilde{\omega}(\|u\|)\|u\|}
<+\infty$,
then
\begin{equation*}
\frac{1}{[\tilde{\omega}(\|u\|)\|u\|]^p}\int_0^TF(t,u(t))dt\to-\infty\quad
\text{as }\|u\|\to\infty.
\end{equation*}
\end{itemize}
\end{lemma}

\begin{proof}
Let $A:=\{t\in[0,T]||u(t)|\geq (T^{-1/p}\|u\|)^{1/2}\}$, for
$u(t)\neq0$, by ($H_1$) and the H\"older's inequality, we
have
\begin{align*}
& \|\nabla F(t,u(t))\|_{L^1}\\
&\leq \int_0^T[f(t)(\omega(|u(t)|)|u(t)|)^{p-1}+g(t)]dt\\
&\leq \|f\|_{L^2}\Big[\int_0^T[\omega(|u(t)|)|u(t)|]^{2(p-1)}dt\Big]^{1/2}+\|g\|_{L^1}\\
&\leq \|f\|_{L^2}\Big[\int_A[\omega(|u(t)|)|u(t)|]^{2(p-1)}dt
+\int_{[0,T]\backslash A}[\omega(|u(t)|)|u(t)|]^{2(p-1)}dt\Big]^{1/2}\hskip-.3cm+\|g\|_{L^1}\\
&\leq \|f\|_{L^2}\Big[\int_0^T\left(\omega((T^{-1/p}\|u\|)^{1/2})|u(t)|\right)^{2(p-1)}dt\\
&\quad +(\sup_{s>0}\omega(s))^{2(p-1)}\int_0^T(T^{-1/p}\|u\|)^{p-1}dt\Big]^{1/2}+\|g\|_{L^1}\\
&\leq \|f\|_{L^2}\left[C_1\left(\omega((T^{-1/p}\|u\|)^{1/2})\|u\|\right)^{2(p-1)}
+C_2\|u\|^{p-1}\right]^{1/2}+\|g\|_{L^1}\\
&\leq \|f\|_{L^2}\left[C_3\left(\omega^2((T^{-1/p}\|u\|)^{1/2})\|u\|^2
+\|u\|\right)\right]^{\frac{p-1}{2}}+\|g\|_{L^1}.
\end{align*}
Take
$\tilde{\omega}(t):=\big[C_3\big(\omega^2((T^{-1/p}t)^{1/2})+\frac{1}{t}\big)
\big]^{1/2},t>0$,
then
\begin{equation*}
\|\nabla
F(t,u(t))\|_{L^1}\leq\|f\|_{L^2}[\tilde{\omega}(\|u\|)\|u\|]^{p-1}+\|g\|_{L^1}.
\end{equation*}
Obviously, $\tilde{\omega}(t)$ satisfies (a) due to the properties
of $\omega(t)$.

Next, we come to check condition (c). Note that
$\liminf_{t\to\infty} \omega(t)/\omega(t^\gamma)>0$
for $\gamma\in(0,1)$, we define
\[
\xi:=\liminf_{t\to\infty}\frac{\omega^2(T^{-1/p}t)}{\omega^2((T^{-1/p}t)
^{1/2})}>0,
\]
if (H2) holds, for any $M>0$, we get
\begin{equation}\label{e2.2}
\int_0^TF(t,u(t))dt\leq-M[\omega(|u(t)|)|u(t)|]^p+C_4,
\end{equation}
which implies that for $\bar{u}\neq0$
\begin{equation}\label{e2.3}
\begin{aligned}
\frac{\int_0^TF(t,\bar{u})dt}{\left[\tilde{\omega}(\|\bar{u}\|)
 \|\bar{u}\|\right]^p}
&\leq \frac{-M\big[\omega(|\bar{u}|)|\bar{u}|\big]^p+C_4}
{\big[C_3\left(\omega^2\left((T^{-1/p}\|\bar{u}\|)^{1/2}\right)
  \|\bar{u}\|^2+\|\bar{u}\|\right)\big]^{p/2}}\\
&= \frac{-M\left[\omega(T^{-1/p}\|\bar{u}\|)T^{-1/p}\|\bar{u}\|
  \right]^p+C_4}{\left[C_3\left(\omega^2
  \left((T^{-1/p}\|\bar{u}\|)^{1/2}\right)\|\bar{u}\|^2+\|\bar{u}\|
  \right)\right]^{p/2}}.
\end{aligned}
\end{equation}
By the definition of $\xi$, there exists $R>0$ such that
\begin{equation}\label{e2.4}
\frac{\omega^2(T^{-1/p}t)t^2}{\omega^2\left((T^{-1/p}t)^{1/2}\right)t^2+t}
\geq\frac{\xi}{2}\quad\text{as }t\geq R.
\end{equation}
Therefore,
\begin{equation}\label{e2.5}
\begin{aligned}
& \frac{1}{[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p}\int_0^TF(t,\bar{u})dt
\\
&\leq \frac{-MT^{-1}\big[\frac{\xi}{2}\left(\omega^2((T^{-1/p}\|\bar{u}\|)^{1/2})\|\bar{u}\|^2
+\|\bar{u}\|\right)\big]^{p/2}+C_4}
{\left[C_3\left(\omega^2((T^{-1/p}\|\bar{u}\|)^{1/2})\|\bar{u}\|^2+\|\bar{u}\|\right)\right]^{p/2}}\\
&\leq -C_5M\quad\text{as }\|\bar{u}\|\to\infty,
\end{aligned}
\end{equation}
condition (c) holds.

Finally, we show that (d) is true. From Lemma \ref{lemma2.2}, for
all $\bar{u}\neq0$, we have
\[
\mathop{\rm meas}\big\{t\in[0,T]|\omega(|\bar{u}|)|\bar{u}|
<m_\beta\omega(|\bar{u}|)\|\bar{u}\|=
m_\beta\omega(T^{-1/p}\|\bar{u}\|)\|\bar{u}\|\big\}<\beta.
\]
Consequently, for all $\bar{u}\neq0$, let
$B:=\{t\in[0,T]|\omega(|\bar{u}|)|\bar{u}|\geq
m_\beta\omega(T^{-1/p}\|\bar{u}\|)\|\bar{u}\|\}$, then we have
meas($[0,T]\backslash B)<\beta$. \vskip.2cm

By (H2*)(3) and Lemma \ref{lemma2.1}, there exists subset
$E_\delta$ of $E$ with meas($E\backslash E_\delta)<\delta$ such that
\begin{equation}\label{e2.6}
\frac{F(t,x)}{[\omega(|x|)|x|]^p}\to-\infty\quad\text{as
}|x|\to\infty
\end{equation}
uniformly for all $t\in E_\delta$. Then, we find
\begin{equation}\label{e2.7}
\mathop{\rm meas}(B\cap
E_\delta)\geq\mathop{\rm meas}(E_\delta)-\mathop{\rm meas}([0,T]\backslash
B)\geq\mathop{\rm meas}(E)-\delta-\beta>0
\end{equation}
for $\delta$ and $\beta$ small enough. By \eqref{e2.6}, for every
$\eta>0$, there exists $L>0$ such that
\begin{equation*}
\frac{F(t,x)}{[\omega(|x|)|x|]^p}\leq-\eta
\end{equation*}
for all $|x|\geq L$ and a.e. $t\in E_\delta$. Furthermore, applying
assumption ($H_2^*$)(2) yields
\begin{equation*}
F(t,x)\leq-\eta[\omega(|x|)|x|]^p+r^*(t)
\end{equation*}
for all $x\in \mathbb{R^N}$ and a.e. $t\in E_\delta$, where
$r^*(t):=(\sup_{s>0}\omega(s))^pL^pr(t)$. Noting the definition of
$\tilde{\omega}(t)$,
$0<\liminf_{t\to\infty}\frac{\omega(t)}{\omega(t^\gamma)}
\leq\limsup_{t\to\infty}\frac{\omega(t)}{\omega(t^\gamma)}<+\infty$
and assumption ($H_2^*$)(1)(iv), we deduce that there exist
$C_6,C_7>0$ such that
\begin{align*}
C_6\omega(T^{-1/p}\|\bar{u}\|)\|\bar{u}\|
&\leq \tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|\\
&=[C_3(\omega^2((T^{-1/p}\|\bar{u}\|)^{1/2})\|\bar{u}\|^2+\|\bar{u}\|)]^{1/2}\\
&\leq C_7\omega(T^{-1/p}\|\bar{u}\|)\|\bar{u}\|\quad\text{as
}\|\bar{u}\|\to\infty,
\end{align*}
which implies
\begin{equation} \label{e2.8}
\begin{aligned}
\int_{B\cap E_\delta}F(t,\bar{u})dt&\leq -\eta\int_{B\cap
E_\delta}[\omega(|\bar{u}|)|\bar{u}|]^pdt+\int_{B\cap
E_\delta}r^*(t)dt\\ &\leq -\eta\int_{B\cap
E_\delta}[\omega(T^{-1/p}\|\bar{u}\|)m_\beta\|\bar{u}\|]^pdt+\int_{B\cap
E_\delta}r^*(t)dt\\
 &\leq -\eta\int_{B\cap
E_\delta}\frac{{m_\beta}^p}{C_7}(\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|)^pdt+\int_{B\cap
E_\delta}r^*(t)dt\\
&\leq -\eta
C_8[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p+\int_{B\cap
E_\delta}\hskip-.3cm r^*(t)dt\quad\text{as
}\|\bar{u}\|\to\infty,
\end{aligned}
\end{equation}
and
\begin{equation}\label{e2.9}
\begin{aligned}
\int_{[0,T]\backslash(B\cap
E_\delta)}F(t,\bar{u})dt
&\leq \int_{[0,T]\backslash(B\cap
E_\delta)}r(t)[\omega(|\bar{u}|)|\bar{u}|]^pdt\\
&\leq \int_0^Tr(t)[\omega(T^{-1/p}\|\bar{u}\|)T^{-1/p}\|\bar{u}\|]^pdt\\
&\leq \int_0^T\frac{T^{-1}}{C_6}(\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|)^pr(t)dt\\
&\leq C_9[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p\int_0^Tr(t)dt\quad\text{as
}\|\bar{u}\|\to\infty.
\end{aligned}
\end{equation}
So, it follows from \eqref{e2.8} and \eqref{e2.9} that
\begin{equation*}
\limsup_{\|\bar{u}\|\to\infty}
\frac{1}{[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p}\int_0^TF(t,\bar{u})dt\leq-\eta
C_8+C_9\int_0^Tr(t)dt,
\end{equation*}
which implies that
\begin{equation}\label{e2.10}
\frac{1}{[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p}
\int_0^TF(t,\bar{u})dt\to-\infty
\quad\text{as }\|\bar{u}\|\to\infty.
\end{equation}
On the other hand, from (b) and the assumptions of (d), we get
\begin{equation} \label{e2.11}
\begin{aligned}
\big|\int_0^TF(t,u(t))dt-\int_0^TF(t,\bar{u})dt\big|
&\leq \int_0^T\int_0^1|\nabla
F(t,\bar{u}+s\tilde{u})||\tilde{u}|\,ds\,dt\\
&\leq \|\tilde{u}\|_\infty[\|f\|_{L^2}(\tilde{\omega}(\|u\|)\|u\|)^{p-1}
  +\|g\|_{L^1}]\\
&\leq C_{10}[\tilde{\omega}(\|u\|)\|u\|]^{p-1}\|\tilde{u}\|
  +C_{11}\|\tilde{u}\|.
\end{aligned}
\end{equation}
As a consequence, note
$\limsup_{\|u\|\to\infty}\frac{\|\tilde{u}\|}{\tilde{\omega}(\|u\|)\|u\|}
<+\infty$,
we then have
\begin{equation}\label{e2.12}
C_{12}:=\limsup_{\|u\|\to\infty}\Big|\frac{1}{[\tilde{\omega}(\|u\|)\|u\|]^p}
\Big[\int_0^TF(t,u(t))dt-\int_0^TF(t,\bar{u})dt\Big]\Big|<+\infty.
\end{equation}
In addition, for $\|u\|\to\infty$, one knows
\begin{equation}\label{e2.13}
1=\frac{\|u\|}{\|u\|}=\frac{\|\bar{u}\|+\|\tilde{u}\|}{\|u\|}
=\frac{\|\bar{u}\|}{\|u\|}+\frac{\|\tilde{u}\|}{\tilde{\omega}(\|u\|)\|u\|}
 \cdot\tilde{\omega}(\|u\|)
=\frac{\|\bar{u}\|}{\|u\|}.
\end{equation}
This, in conjunction with \eqref{e2.10}-\eqref{e2.12}, gives
\begin{equation} \label{e2.14}
\begin{aligned}
& \limsup_{\|u\|\to\infty}\frac{1}{[\tilde{\omega}(\|u\|)\|u\|]^p}\int_0^T
 F(t,u(t))dt\\
&\leq\limsup_{\|u\|\to\infty}\frac{1}{[\tilde{\omega}(\|u\|)\|u\|]^p}\int_0^TF(t,\bar{u})dt+C_{12}\\
&=\limsup_{\|\bar{u}\|\to\infty}\frac{1}{[\tilde{\omega}(\|\bar{u}\|)\|\bar{u}\|]^p}\int_0^TF(t,\bar{u})dt+C_{12}
\to-\infty,
\end{aligned}
\end{equation}
which completes the proof.
\end{proof}

\section{Proofs of theorems}

Now, we give the proofs of the main results.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
First, we prove that $\varphi$ satisfies the (PS) condition. Suppose
that $\{u_n\}$ is a (PS)
sequence for $\varphi$, that is, $\varphi'(u_n)\to0$ as
$n\to\infty$ and $\{\varphi(u_n)\}$ is bounded. It follows
from Wirtinger's inequality that
\begin{equation}\label{e3.1}
\|\dot{u}_n\|_{L^p}\leq\|\tilde{u}_n\|\leq(C_0+1)^{1/p}\|\dot{u}_n\|_{L^p}
\end{equation}
for all $n$. By virtue of the properties of $\tilde{\omega}(t)$, one
has
\begin{equation}\label{e3.2}
\begin{aligned}
\tilde{\omega}(\|\bar{u}+\tilde{u}\|)\leq\min\{\tilde{\omega}(\|\bar{u}\|),\tilde{\omega}(\|\tilde{u}\|)\},
\end{aligned}
\end{equation}
which implies
\begin{equation} \label{e3.3}
\begin{aligned}
& \big|\int_0^T(\nabla
F(t,u_n(t)),\tilde{u}_n(t))dt\big|\\
&\leq\|\tilde{u}_n\|_\infty\left[\|f\|_{L^2}
(\tilde{\omega}(\|u_n\|)\|u_n\|)^{p-1}+\|g\|_{L^1}\right]\\
&\leq \|\tilde{u}_n\|_\infty\left[\|f\|_{L^2}
(\tilde{\omega}(\|\bar{u}_n+\tilde{u}_n\|)\|\bar{u}_n+\tilde{u}_n\|)^{p-1}+\|g\|_{L^1}\right]\\
&\leq \|\tilde{u}_n\|_\infty\left[\|f\|_{L^2}
(\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|
+\tilde{\omega}(\|\tilde{u}_n\|)\|\tilde{u}_n\|)^{p-1}+\|g\|_{L^1}\right]\\
&\leq C_{13}\|\tilde{u}_n\|\left[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|\right]^{p-1}
 +C_{13}\|\tilde{u}_n\|\left[\tilde{\omega}(\|\tilde{u}_n\|)\|\tilde{u}_n\|\right]^{p-1}\hskip-.2cm+C_{14}\|\tilde{u}_n\|
\end{aligned}
\end{equation}
for all $n$. Thus, by \eqref{e3.1} and \eqref{e3.3}, we get
\begin{equation} \label{e3.4}
\begin{aligned}
\|\tilde{u}_n\|
&\geq (\varphi'(u_n),\tilde{u}_n)\\
&=\int_0^T|\dot{u}_n(t)|^pdt+\int_0^T(\nabla
F(t,u_n(t)),\tilde{u}_n(t))dt\\
&\geq C_{15}\|\tilde{u}_n\|^p-C_{13}\|\tilde{u}_n\|\left[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|\right]^{p-1}\\
&\quad -C_{13}\|\tilde{u}_n\|\left[\tilde{\omega}(\|\tilde{u}_n\|)
 \|\tilde{u}_n\|\right]^{p-1}-C_{14}\|\tilde{u}_n\|.
\end{aligned}
\end{equation}
Assume that $\{\|\tilde{u}_n\|\}$ is unbounded; that is,
$\|\tilde{u}_n\|\to\infty$ as $n\to\infty$. Since
$\tilde{\omega}(t)\to0$ as $t\to\infty$, it follows
from \eqref{e3.4} that we can find a constant $C_{16}>0$ such that
\begin{equation}\label{e3.5}
C_{16}\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|\geq\|\tilde{u}_n\|,
\end{equation}
which implies
\begin{equation*}
 [C_3(\omega^2((T^{-1/p}\|\bar{u}_n\|)^{1/2})\|\bar{u}_n\|^2+\|\bar{u}_n\|)]^{1/2}
 =\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|\to\infty
\quad\text{as }n\to\infty.
\end{equation*}
Since $\omega$ is bounded, this leads to
\begin{equation}\label{e3.6}
\|\bar{u}_n\|\to\infty\quad\text{as }n\to\infty.
\end{equation}
On the other hand,
$\|\bar{u}_n+s\tilde{u}_n\|\geq\|\bar{u}_n\|,s\in[0,1]$, by Lemma
\ref{lemma2.3} (b) and \eqref{e3.5}, we see that
\begin{equation} \label{e3.7}
\begin{aligned}
& \int_0^T[F(t,u_n(t))-F(t,\bar{u}_n)]dt\\
&\leq \big|\int_0^T\int_0^1(\nabla
F(t,\bar{u}_n+s\tilde{u}_n(t)),\tilde{u}_n(t))\,ds\,dt\big|\\
&\leq \|\tilde{u}_n\|_\infty\int_0^T\int_0^1|\nabla
F(t,\bar{u}_n+s\tilde{u}_n(t))|\,ds\,dt\\
&\leq \|f\|_{L^2}\left[\tilde{\omega}(\|\bar{u}_n+s\tilde{u}_n\|)\|\bar{u}_n+s\tilde{u}_n\|\right]^{p-1}
\|\tilde{u}_n\|_\infty+\|g\|_{L^1}\|\tilde{u}_n\|_\infty\\
&\leq \|f\|_{L^2}[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|+\tilde{\omega}(\|\bar{u}_n\|)\|\tilde{u}_n\|]^{p-1}
\|\tilde{u}_n\|_\infty+\|g\|_{L^1}\|\tilde{u}_n\|_\infty\\
&\leq C_{17}\|\tilde{u}_n\|[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^{p-1}
+C_{17}\|\tilde{u}_n\|[\tilde{\omega}(\|\bar{u}_n\|)\|\tilde{u}_n\|]^{p-1}+C_{18}\|\tilde{u}_n\|\\
&\leq C_{19}[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p
+C_{20}[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p[\tilde{\omega}(\|\bar{u}_n\|]^{p-1}
+C_{21}\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|,
\end{aligned}
\end{equation}
which implies that
\begin{equation} \label{e3.8}
\begin{aligned}
\varphi(u_n)
&= \frac{1}{p}\|\dot{u}_n\|_{L^p}^p+\int_0^T[F(t,u_n(t))-F(t,\bar{u}_n)]dt+\int_0^TF(t,\bar{u}_n)dt\\
&\leq C_{22}[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p
+C_{20}[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p[\tilde{\omega}(\|\bar{u}_n\|]^{p-1}\\
&\quad +C_{21}\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|+\int_0^TF(t,\bar{u}_n)dt\\
&\leq [\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p\Big[C_{22}+C_{20}[\tilde{\omega}(\|\bar{u}_n\|]^{p-1}
+\frac{C_{21}}{[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^{p-1}}\\
&\quad +\frac{1}{[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p}
\int_0^TF(t,\bar{u}_n)dt\Big]
\to -\infty\quad\text{as }\|\bar{u}_n\|\to\infty.
\end{aligned}
\end{equation}
This contradicts the boundedness of $\varphi(u_n)$. So
\begin{equation}\label{e3.9}
\{\|\tilde{u}_n\|\}\text{ is bounded}.
\end{equation}
Suppose that $\{\|\bar{u}_n\|\}$ is unbounded and
$\{\|\tilde{u}_n\|\}$ is bounded. With the similar manner above, we
deduce that
\begin{equation} \label{e3.10}
\begin{aligned}
\varphi(u_n)
&=\frac{1}{p}\|\dot{u}_n\|_{L^p}^p+\int_0^T[F(t,u_n(t))-F(t,\bar{u}_n)]dt
+\int_0^TF(t,\bar{u}_n)dt\\
&\leq C_{23}\|\tilde{u}_n\|^p+C_{17}\|\tilde{u}_n\|[\tilde{\omega}(\|\bar{u}_n\|)\|\tilde{u}_n\|]^{p-1}
+C_{18}\|\tilde{u}_n\|+\int_0^TF(t,\bar{u}_n)dt\\
&\leq C_{24}+C_{25}[\tilde{\omega}(\|\bar{u}_n\|)\|\tilde{u}_n\|]^{p-1}+C_{26}[\tilde{\omega}(\|\bar{u}_n\|)]^{p-1}
+\int_0^TF(t,\bar{u}_n)dt\\
&\leq [\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p\Big[\frac{C_{24}}{[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p}
+\frac{C_{25}}{\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|}
+\frac{C_{26}}{\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|^p}\\
&\quad +\frac{1}{[\tilde{\omega}(\|\bar{u}_n\|)\|\bar{u}_n\|]^p}
 \int_0^TF(t,\bar{u}_n)dt\Big]
\to -\infty\quad\text{as }\|\bar{u}_n\|\to\infty,
\end{aligned}
\end{equation}
 which also contradicts the boundedness of $\varphi(u_n)$. Then
\begin{equation}\label{e3.11}
\{\|\bar{u}_n\|\}\text{ is also bounded}.
\end{equation}
From \eqref{e3.9} and \eqref{e3.11}, we have $\{\|u_n\|\}$ is bounded,
thus $\varphi$ satisfies the (PS) condition.

Since $W_T^{1,p}=\mathbb{R^N}\oplus \tilde{W}_T^{1,p}$, where
$\tilde{W}_T^{1,p}:=\{u\in W_T^{1,p}|\int_0^Tu(t)dt=0\}$. Next, we
shall prove that
\begin{equation}\label{e3.12}
\varphi(u)\to+\infty\quad\text{as }\|u\|\to\infty\text{ in }\tilde{W}_T^{1,p}.
\end{equation}
In fact, since $\tilde{\omega}(t)\to0$ as
$t\to\infty$, then there exists $A>0$ such that
$\tilde{\omega}(A)\leq\frac{1}{2pC_{29}}$. In a similar way to
\eqref{e3.7}, we have
\begin{align*}
&\big|\int_0^T[F(t,u(t))-F(t,A)]dt\big|\\
&\leq C_{17}\|\tilde{u}\|[\tilde{\omega}(A)A]^{p-1}
+C_{17}\|\tilde{u}\|[\tilde{\omega}(A)\|\tilde{u}\|]^{p-1}+C_{18}\|\tilde{u}\|\\
&\leq C_{27}\|\tilde{u}\|+C_{17}\tilde{\omega}(A)\|\tilde{u}\|^p\\
&\leq C_{28}\|\dot{u}\|_{L^p}+C_{29}\tilde{\omega}(A)\|\dot{u}\|_{L^p}^p\\
&\leq C_{28}\|\dot{u}\|_{L^p}+\frac{1}{2p}\|\dot{u}\|_{L^p}^p,
\end{align*}
which implies
\begin{equation} \label{e3.13}
\begin{aligned}
\varphi(u)
&= \frac{1}{p}\|\dot{u}\|_{L^p}^p+\int_0^T[F(t,u(t))-F(t,A)]dt
  -\int_0^TF(t,A)dt\\
&\geq  \frac{1}{p}\|\dot{u}\|_{L^p}^p-\frac{1}{2p}\|\dot{u}\|_{L^p}^p
  -C_{28}\|\dot{u}\|_{L^p}-\int_0^TF(t,A)dt
\end{aligned}
\end{equation}
for all $u\in \tilde{W}_T^{1,p}$. By Wirtinger's inequality, one has
\begin{equation*}
\|u\|\to\infty\Leftrightarrow\|\dot{u}\|_{L^p}\to\infty\quad
\text{on }\tilde{W}_T^{1,p}.
\end{equation*}
Hence, \eqref{e3.12} follows from \eqref{e3.13}.
On the other hand, by (H2),
\begin{equation}\label{e3.14}
\varphi(u)\to-\infty\quad\text{as }|u|\to\infty\quad\text{in }\mathbb{R^N}.
\end{equation}
Combine \eqref{e3.12} and \eqref{e3.14}, applying saddle point
theorem, then problem \eqref{e1.1} has at least one solution in
$W_T^{1,p}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
We commence by showing that
$\varphi$ satisfies (PS) condition. Let $\{u_n\}$ be a sequence in
$W_T^{1,p}$ such that $\{\varphi(u_n)\}$ is bounded and
$\varphi'(u_n)\to0$ as $n\to\infty$. If $\{u_n\}$ is
unbounded, without loss of generality, we may assume that
$\|u_n\|\to\infty$ as $n\to\infty$. It follows from
Lemma \ref{lemma2.3} (b) that
\begin{equation} \label{e3.15}
\begin{aligned}
\big|\int_0^T(\nabla F(t,u_n(t)),\tilde{u}_n(t))dt\big|
&\leq \|\tilde{u}_n\|_\infty[\|f\|_{L^2}[\tilde{\omega}(\|u_n\|)\|u_n\|
  ]^{p-1}+\|G\|_{L^1}]\\
&\leq C_{30}[\tilde{\omega}(\|u_n\|)\|u_n\|]^{p-1}\|\tilde{u}_n\|
 +C_{31}\|\tilde{u}_n\|.
\end{aligned}
\end{equation}
Hence, we have
\begin{align*}
\|\tilde{u}_n\|
&\geq  (\varphi'(u_n),\tilde{u}_n)\\
&= \int_0^T|\dot{u}_n(t)|^pdt+\int_0^T(\nabla
F(t,u_n(t)),\tilde{u}_n(t))dt\\
&\geq  C_{32}\|\tilde{u}_n\|^p-C_{30}[\tilde{\omega}(\|u_n\|)\|u_n\|]^{p-1}
 \|\tilde{u}_n\|-C_{31}\|\tilde{u}_n\|,
\end{align*}
which implies
\begin{equation}\label{e3.16}
\limsup_{\|u_n\|\to\infty}\frac{\|\tilde{u}_n\|}{\tilde{\omega}(\|u_n\|)\|u_n\|}<+\infty.
\end{equation}
Therefore, by Lemma \ref{lemma2.3} (d), one has
\begin{equation}\label{e3.17}
\frac{1}{[\tilde{\omega}(\|u_n\|)\|u_n\|]^p}\int_0^T
F(t,u_n(t))dt\to-\infty\quad\text{as }\|u_n\|\to\infty.
\end{equation}
However, by the boundedness of $\varphi(u_n)$ and \eqref{e3.16}, we get
\begin{equation}\label{e3.18}
\begin{aligned}
\big|\frac{1}{[\tilde{\omega}(\|u_n\|)\|u_n\|]^p}\int_0^TF(t,u_n(t))dt
\big|
&=\big|\frac{\varphi(u_n)}{[\tilde{\omega}(\|u_n\|)\|u_n\|]^p}
-\frac{\frac{1}{p}\int_0^T|\dot{u}_n(t)|^pdt}{[\tilde{\omega}(\|u_n\|)
 \|u_n\|]^p}\big|\\
&\leq \frac{\frac{1}{p}\int_0^T|\dot{u}_n(t)|^pdt}{[\tilde{\omega}(\|u_n\|)
 \|u_n\|]^p}\\
&\leq\frac{C_{33}\|\tilde{u}_n\|^p}{[\tilde{\omega}(\|u_n\|)\|u_n\|]^p}
<+\infty,
\end{aligned}
\end{equation}
which contradicts \eqref{e3.17}. So, $\varphi$ satisfies the (PS)
condition.

As in the proof of Theorem \ref{thm1.1}, we can obtain
\begin{equation*}
\varphi(u)\to+\infty\quad\text{as
}\|u\|\to\infty\text{ in }\tilde{W}_T^{1,p}.
\end{equation*}
 Furthermore, from \eqref{e2.10}, it is easy to see that
\begin{equation*}
\varphi(u)\to-\infty\quad\text{as }
 |u|\to\infty\quad\text{in }\mathbb{R^N}.
\end{equation*}
Thus, using the saddle point theorem, problem \eqref{e1.1} has at
least one solution in $W_T^{1,p}$.
\end{proof}

\subsection*{Acknowledgments}
 This Project was Supported by
Foundation of Major Project of Science and Technology of Chinese
Education Ministry, SRFDP of Higher Education and NSF of Education
Committee of Jiangsu Province and Foundation of Nanjing University
of Information Science $\&$ Technology. The authors would like to
thank the anonymous referee for his/her valuable suggestions.


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\end{document}