\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 166, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/166\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for semilinear
  differential equations with subquadratic potentials}

\author[C. Li, M. Wang, Z. Xiao \hfil EJDE-2013/166\hfilneg]
{Chengyue Li, Mengmeng Wang, Zhiwei Xiao} 

\address{Chengyue Li \newline
Department of Mathematics, Minzu University of
China, Beijing 100081, China}
\email{cunlcy@163.com}

\address{Mengmeng Wang \newline
Department of Mathematics, Minzu University of
China, Beijing 100081, China}
\email{wang-meng-1989@163.com}

\address{Zhiwei Xiao \newline
Department of Mathematics, Minzu University of
China, Beijing 100081, China}
\email{paopao\_0811@126.com}

\thanks{Submitted April 2, 2013. Published July 21, 2013.}
\thanks{Partially supported by the China Scholarship Council (CSC)}
\subjclass[2000]{58E05, 34C37, 70H05}
\keywords{Subquadratic potentials; Clark theorem; critical points;
\hfill\break\indent  Hamiltonian systems}

\begin{abstract}
 Existence and multiplicity of nontrivial solutions of a class of semilinear
 differential equations with subquadratic potentials are studied using
 Clark's theorem. As an application of the above results,
 periodic solutions for a second Hamiltonian systems are studied.
\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}

In physical, chemical and biological sciences,
many models can be set as  equation of the form
\begin{equation}
  \mathcal{L} u=V_{u}( t,u ),\quad t\in \mathbb{R}, \label{eP}
\end{equation}
where $\mathcal{L} $ is a linear
differential operator which is self-adjoint and positive in some
suitable space, and the potential
 $V(t,q):\mathbb{R}\times {\mathbb{R}^n}\to \mathbb{R}$ 
is a $C^1$-function, with $V_{u}(t,q)=\frac{\partial V}{\partial u}$; 
see for example the references in this article.
If $V$ satisfies
\[
\lim_{u\to\infty }V(t,u)/ | u |^{^2}\leqslant c<\infty ,
\]
then we say that \eqref{eP} is
subquadratic. If $\lim_{|u|\to \infty }V(u)/|u|^2=\infty$, 
then \eqref{eP} is superquadratic. 
Our main goal is to find periodic solutions of \eqref{eP} with 
subquadratic potentials by
variational methods; that is, periodic solutions of \eqref{eP} are
critical points of the corresponding functional in an appropriate
Hilbert space $( X ,\| \cdot  \| )$,
given by
\begin{equation}
 \varphi _{T}(u)=\frac{1}{2}\|u\|^2-J(u),\quad u\in \mathbb{X}, \label{e1}
\end{equation}
where $\langle \mathcal{L} u,u \rangle=\frac{1}{2}\| u \|^2$,
$\langle , \rangle $ denotes the inner product of $X$.
Under some assumptions, we have an abstract result as follows.

\begin{theorem} \label{thm1} \rm
 Let $(X,\|\cdot \|)\subset L^2\equiv L^2([ 0 ,T],\mathbb{R}^n)$
be a Hilbert space such that
\begin{equation}
  \| u \|_{L^2(0,T)}\leqslant\big( \frac{T}{\pi }\big)^{k}\| u\|,\quad
\forall u\in X \label{e2}
\end{equation}
for some $k\in \mathbb{N}$, and $\{ e_j ( t )\}_{j=1}^{\infty }$ be an
orthogonal sequence in $X$ and
$L^2$ such that $| e_j( t ) |\leqslant 1$, for all $j\geqslant 1$,
$t\in [0,T] $ and
\begin{equation}
 \| e_j(t)\|_{L^2(0,T)}^2\geqslant \mu, \quad
 \|e_j(t))\| ^{^2}\leqslant \mu (\frac{j\pi }{T})^{2k},\quad
\forall j\geqslant 1\label{e3}
\end{equation}
for some $\mu > 0$, the functional $\varphi _{T}(u)$ in $(1)$  is
defined in $X$ such that $J(u)\in \mathbb{C}^1(X,\mathbb{R})$,
$J(0)=0$ and $J'(u)$ is completely continuous. Furthermore, we
assume that $J(u)$ satisfies:
\begin{itemize}
\item[(J1)] $J(u)=J(-u)$ for all $u\in X$,

\item[(J2)] there exist $m>0$ and $b>0$ such that for all $u\in X$,
$T<\pi/\sqrt[2k]{m}$, and $J(u)\leqslant b+\frac{1}{2}m\| u \|_{L^2(0,T)}^2$,


\item[(J3)] there exist $ p\in \mathbb{N}$ and
$M>0$, $\rho >0 $such that $T>p \pi /\sqrt[2k]{M}$, $M>mp^{2k}$, and
\[
  J( u )\geqslant \frac{1}{2}M\| u \|_{L^2(0,T)}^2,\quad
\forall u\in  X \text{ with } \| u \|_{L^{\infty }(0,T) }\leqslant \rho
 \sqrt{p}.
\]
\end{itemize}
Then, for each $T\in(p \pi /\sqrt[2k]{M}, \pi /\sqrt[2k]{m})$,
there exist at least $p$ distinct critical point pairs $( u_{i},-u_{i} )$
 of $\varphi _{T}( u )$ such that
$\varphi _{T}( u_{i} )<0( 1\leqslant i\leqslant p  )$.
\end{theorem}

In section 2, we will give the proof of Theorem \ref{thm1}.
For this, we recall a compactness condition introduced by Palais and Smale.

\textbf{Palais-Smale condition.} Let $X$ be a Banach space and
$\varphi \in \mathbb{C}^1( X,\mathbb{R} )$ is said to
satisfy Palais-Smale condition if any sequence
$\{ u_j  \} \subset X$ such that
$\varphi (u_j )$ is bounded and ${\varphi }'( u_j )\to 0$
possesses a convergent subsequence.

We also need the following theorem.

\begin{theorem}[Clark Theorem \cite{c2}] \label{clarkthm}
Let $X$be a Banach space and $\varphi \in
\mathbb{C}^1(X,\mathbb{R} )$ be even satisfying the
Palais-Smale condition. Suppose that
(i) $\varphi$ is bounded from below; (ii) there exist a closed,
symmetric set $K\subset X$ and $p\in \mathbb{N}$ such that $K$  is
homeomorphism to $S^{p-1}$ by an odd map, and
$\sup\{ \varphi ( x ):x\in K \}< \varphi ( 0 )$.
Then $\varphi$ possesses at least $ p $distinct pairs
 $( u,-u )$ of critical point with corresponding critical values less
than $\varphi ( 0 )$.
\end{theorem}

In section 3, we will apply Theorem \ref{thm1} to Hamiltonian systems and fourth-order
differential equations with bi-even subquadratic
potentials. First, we consider second order
Hamiltonian systems
\begin{equation}
 \ddot{u}( t )+V_{u}( t,u( t ) )=0,\quad t\in \mathbb{R} \label{eHS}
\end{equation}
where $V( t,u )\in \mathbb{C}^1( \mathbb{R}\times \mathbb{R}^n,\mathbb{R} )$
is a $T$-periodic function in the variable $t$,$V( t,0 )\equiv 0$.
 The existence of periodic solutions is
one of the most important problems in the theory of Hamiltonian
systems. In the past thirty years, many authors studied periodic
solutions for Hamiltonian systems via the critical point theory.
Here we only mention some results for subquadratic Hamiltonian
systems. Clarke and Ekeland \cite{c3}
studied a family of convex subquadratic Hamiltonian systems where
$V( t,u )=V( u )$ satisfies
$\lim_{|u|\to \infty }V( u )/|u|^2=0$,
$\lim_{|u|\to 0 }V( u )/|u|^2=\infty $,  and
they used the dual variational method to obtain the first
variational result on periodic solutions having a prescribed minimal
period. Later, Mawhin and Willem \cite{m1}  made an improvement,
they supposed that $V( u )$ is convex, also satisfies
$\lim_{|u|\to \infty }V( u)/|u|^2=0$,
$\lim_{|u|\to 0 }V( u )/|u|^2=\infty $, and proved that there is
a $T_0> 0$ such that for all $T> T_0$, \eqref{eHS} has a periodic solution
with minimal period $T$. Rabinowitz \cite{r1,r2}, Tang \cite{t1} and others 
proved the existence, where authors used the  subquadratic
condition: $uV_{u}( t,u )\leqslant \alpha V( t,u)( 0< \alpha < 2 )$,
which plays an important role. Schechter \cite{s1} assumed
that $V( t,u )$ is subquadratic, and
$2V( t,u)-uV_{u}( t,u )\to -\infty ( | u | \to \infty )$ or
$2V( t,u )-uV_{u}( t,u )\leqslant W( t )$, then he proved that \eqref{eHS}
has one non-constant periodic solution. 
Long \cite{l1} studied this problem for bi-even subquadratic potentials, and get
the existence of one odd $T$-periodic solution. Inspired by the above papers,
 using Theorem \ref{thm1}, we give a multiplicity result for
\eqref{eHS} as follows.

\begin{theorem} \label{thm2}
Let $V( t,u)\in \mathbb{C}^1( \mathbb{R}\times \mathbb{R}^n,\mathbb{R} )$
be $\tau$-periodic in $t$ and satisfy
\begin{itemize}
\item[(V1)] $V( t,u )=V( t,-u )=V( -t,-u )$, for all
$t\in \mathbb{R}$, $u\in \mathbb{R}^n$;

\item[(V2)] there exist $m>0,b>0$ such
that $\tau < 2\pi /\sqrt{m}$ and $V( t,u )\leqslant b+\frac{1}{2}m| u |^2$,
for all $t\in \mathbb{R}$, $u\in \mathbb{R}^n$;

\item[(V3)] there exist $p\in \mathbb{N}$ and constants $M> 0,\rho > 0$
such that $\tau >  2p\pi /\sqrt{M}$, $M> mp^2$ and
\[
V( t,u )\geqslant  \frac{1}{2}M| u |^2,\quad
\forall t\in \mathbb{R},\, | u |\leqslant \rho
\sqrt{p}.
\]
\end{itemize}
Then, for $\tau \in( 2p\pi /\sqrt{M},2\pi /\sqrt{m} )$,\eqref{eHS} has
$p$ distinct pairs $( u( t),-u( t ) )$ of nontrivial odd
$\tau$-periodic solutions.
\end{theorem}

\begin{remark} \label{rmk1} \rm
If $V( t,u )=a( t )W( u )$, $a( t )$ and $W( u )$ are even, then
(V1) holds.
\end{remark}

Note that our method and results in this article are different from
the earlier ones in \cite{l1,m1,r1,r2,s1,t1} and references therein.

As the second application, in the study of formation
of spatial patterns in bistable systems, we consider a
fourth-order differential equation \cite{c1,g1,t2},
\begin{equation}
 u^{( 4 )}( t )-V_{u}( t,u( t ) )=0,\quad  0\leqslant t\leqslant T\label{eFE}
\end{equation}
with the boundary condition $u( 0 )=u( T )={u}''( 0 )={u}''( T )=0$.
For \eqref{eFE}, we also have a result similar to Theorem \ref{thm2}.

\begin{theorem} \label{thm3}
Let $V( t,u )\in \mathbb{C}^1( \mathbb{R}\times \mathbb{R},\mathbb{R} )$
satisfy
\begin{itemize}
\item[(V4)]  $V( t,u ) =V( t,-u )$, for all $t\in \mathbb{R},u\in \mathbb{R}$;

\item[(V5)]there exist $m>0,b>0$ such that $T< \pi /\sqrt[4]{m}$ and
$V( t,u )\leqslant b+\frac{1}{2}m| u |^2$, for all
$t\in \mathbb{R},u\in \mathbb{R}$;

\item[(V6)] there exist $p\in \mathbb{N}$ and constants $M> 0,\rho > 0$
such that $T>  p\pi /\sqrt[4]{M}$,$M> mp^{4}$ and
\[
V( t,u )\geqslant  \frac{1}{2}M| u |^2,\quad
\forall t\in \mathbb{R},\, | u |\leqslant \rho \sqrt{p}.
\]
\end{itemize}
Then, for each $T\in ( p\pi /\sqrt[4]{M},\pi /\sqrt[4]{m} )$,
\eqref{eFE} has at least $p$ distinct pairs $( u( t),-u( t ) )$ of solutions.
\end{theorem}

 To the best of our knowledge, only a few multiplicity results for
fourth-differential equations, similar to Theorem \ref{thm3},  have been reported
in the literature.

\section{Proof of Theorem \ref{thm1}}

\begin{lemma} \label{lem1}
Under the hypotheses of Theorem \ref{thm1}, the functional $\varphi _{T}( u )$
is coercive in $X$, and bounded from below.
\end{lemma}

\begin{proof}  The condition (J2) implies the estimate
\begin{equation}
\varphi _{T}( u )\geqslant \frac{1}{2}\| u \|^2
-\frac{1}{2}m\| u \|_{L^2( 0,T )}^2-b,\label{e4}
\end{equation}
which combined with the inequality \eqref{e2} yields
\begin{equation}
\varphi _{T}( u )\geqslant \frac{1}{2}
\big[ 1-m( \frac{T}{\pi } ) ^{2k}\big]\| u \|^2-b
=\frac{1}{2}B\| u \|^2-b\geqslant -b,\label{e5}
\end{equation}
with $B=1-m( \frac{T}{\pi } ) ^{2k}> 0$. Thus, we conclude
that $\varphi _{T}( u )$ is coercive in $X$, and
bounded from below.
\end{proof}

\begin{lemma} \label{lem2}
Under the hypotheses of Theorem \ref{thm1}, the functional
$\varphi _{T}( u)$ satisfies Palais-Smale condition in
$X$.
\end{lemma}

\begin{proof}
Let $\{u_j\}\subset X $ be such that
$\varphi _{T}(u_j )$ is bounded and ${\varphi _{T}}'( u_j )\to 0( j\to \infty  )$.
Then by Lemma \ref{lem1},
\begin{equation}
\| u_j \|^2\leqslant \frac{2}{B}( \varphi_{T} ( u_j )+b )\label{e6}
\end{equation}
which implies that $\{ u_j \}$ is bounded, so we may assume,
by passing to a subsequence if necessary, that
\begin{equation}
u_j\rightharpoonup u\quad \text{weakly in } X.\label{e7}
\end{equation}
As we know
\begin{equation}
{\varphi _{T}}'( u_j )u=\langle u_j ,u\rangle-{J}'( u_j )u\label{e8}
\end{equation}
letting $j\to \infty $, by the completely continuousness of
$J'$, we have
\begin{equation}
0=\| u \|^2-{J}'( u )u\,.\label{e9}
\end{equation}
Since
$|{\varphi _{T}}'( u_j )u_j  |\leqslant\|{\varphi _{T}}'( u_j ) \|\|u_j  \|\to 0$,
using \eqref{e9}, we obtain
\begin{equation}
\|u_j  \|^2={\varphi _{T}}'( u_j )u_j+{J}'( u_j )u_j\to {J}'( u)u
=\| u \|^2\label{e10}
\end{equation}
Thus, we conclude that $u_j\to u$ in $X$. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
For $p\in \mathbb{N}$ and $\rho > 0$ defined in (J3), let
the subset $K$ of $X$ as follows
\begin{equation}
K=\{ \lambda _{1}e_{1}( t ) +\lambda _{2}e_{2}( t
)+\dots+\lambda _{p}e_{p}( t ):\lambda_1,\lambda _2,\dots,\lambda _p
\in \mathbb{R},\sum_{k=1}^{p}\lambda_k^2=\rho ^2\}\label{e11}
\end{equation}
We know the map
\begin{equation}
\lambda _{1}e_{1}( t ) +\lambda _{2}e_{2}( t )+\dots
+\lambda _{p}e_{p}( t )\to ( -\frac{\lambda _{1}}{\rho } ,
-\frac{\lambda _{2}}{\rho },\dots,-\frac{\lambda _{p}}{\rho })
\label{e12}
\end{equation}
is an odd homeomorphism from $K$ to
$S^{p-1}$. For all $u( t )=\lambda _{1}e_{1}(
t ) +\lambda _{2}e_{2}( t )+\dots+\lambda
_{p}e_{p}( t )\in K$, we have the estimate
\begin{equation}
| u( t ) |^2\leqslant( \lambda _{1} ^2+\lambda _{2}^2+\dots+\lambda
_{p}^2 )(| e_{1}( t ) |^2+|e_{2}( t ) |^2+\dots+|
e_{p}( t ) |^2)\leqslant p\rho
^2.\label{e13}
\end{equation}
Combining \eqref{e13} and (J3) with
\eqref{e3} shows that
\begin{equation}
\begin{aligned}
\varphi _{T}( u )
&\leqslant \frac{1}{2}\| u \|^2-\frac{1}{2}M\| u \|_{L^2(0,T)}^2 \\
&=\frac{1}{2}\sum _{j=1}^{p}\lambda _j^2\| e_j( t ) \|^2-\frac{1}{2}
 M\sum _{j=1}^{p}\lambda _j^2\|
e_j( t ) \|_{L^2(0,T)}^2 \\
&\leqslant \frac{\mu }{2}\sum _{j=1}^{p}\lambda
_j^2[ ( \frac{j\pi }{T} )^{2k}-M].
\label{e14}
\end{aligned}
\end{equation}
Since $T> p\pi /\sqrt[2k]{M}$, we have $0<\frac{j\pi }{T}<\sqrt[2k]{M}$,
for $1\leqslant j\leqslant p$. Therefore, we obtain
\begin{equation}
\varphi _{T}( u )\leqslant
\frac{\mu }{2}\rho ^2( ( \frac{p\pi }{T} )^{2k}-M )<0,\quad
\forall u\in K\label{e15}
\end{equation}
which implies $sup\{ \varphi _{T}( u ):u\in K \}<0= \varphi _{T}( 0 )$.
Thus, Lemma \ref{lem1}, Lemma \ref{lem2} and Clark Theorem imply
that $\varphi _{T}( u )$ possesses at least $p$
distinct pairs $( u_{i},-u_{i} )$ of critical points
such that $\varphi _{T}( u_{i} )< 0$ clearly $u_{i}\neq
0( 1\leqslant i\leqslant p )$.
\end{proof}

\begin{corollary} \label{coro1}
In Theorem \ref{thm1}, for any $T> 0$, if conditions {\rm (J2)-(J3)}
are replaced by
\begin{itemize}
\item[(J2')] $\lim_{| u |\to \infty }V( t,u)/| u |^2=0$ uniformly in
$t\in \mathbb{R}$,
\item[(J3')] $\lim_{| u |\to 0 }V( t,u )/| u |^2=\infty $ uniformly in
$t\in \mathbb{R}$,
\end{itemize}
then the functional
$\varphi _{T}( u )$ has infinitely many distinct pairs
$( -u,u )$ of critical points.
\end{corollary}

\begin{proof}
For any fixed $p\in\mathbb{N}$, by (J2') and (J3'),
we may take $m$ sufficiently small and $M$ large
enough such that
\begin{equation}
 0<m<( \frac{\pi}{T} )^{2k},\quad
M>( \frac{p\pi}{T} )^{2k}, \quad  M>mp^{2k}, \label{e16}
\end{equation}
thus (J2)-(J3) are all satisfied, and
$T\in ( p\pi /\sqrt[2k]{M},\pi /\sqrt[2k]{m} )$.
By Theorem \ref{thm1}, the functional $\varphi _{T}( u )$ has
at least $p$ distinct pairs $(u,-u)$ of critical points.
Since $p$ is arbitrary, there exist infinitely many distinct
pairs $( u,-u )$  of critical points of $\varphi_T( u )$.
\end{proof}

\section{Applications of Theorem \ref{thm1}}
The first application is for to Hamiltonian systems. 
To prove Theorem \ref{thm2}, we study the related boundary value problem
\begin{equation}
\begin{gathered}
\ddot{u}( t )+V_{u}( t,u( t ) )=0,\quad 0< x< T,\\
u( 0 )=u( T )=0,
\end{gathered}\label{e17}
\end{equation}
with $T=\tau/2$. For a solution $u(t )$ of \eqref{e17}, we define
\begin{equation}
\bar{u}( t )= \begin{cases}
u( t )  & 0\leqslant t\leqslant T,\\
-u( -t )&-T\leqslant t\leqslant 0.
\end{cases}\label{e18}
\end{equation}
For any $t\in [ -T,0 ]$, by (V1), we see that
\begin{align*}
{(\bar{u} )}''( t )+V_{u}( t,\bar{u}( t ) )
&=-\ddot{u}( -t ) + V_{u}( t,-u( -t ) ) \\
&=-[ \ddot{u}( -t )-V_{u}( t,-u( -t) ) ] \\
&=-[ \ddot{u}( -t ) +V_{u}( t,-u( -t ) )]=0.
\end{align*}
Hence, $\bar{u}= \bar{u}( t )$ is a solution of \eqref{eHS} over 
$[-T,T ]$,and its $2T$-periodic extension over $\mathbb{R}$,
still denoted by $\bar{u}= \bar{u}( t )$, is an odd
$\tau$-periodic solution of \eqref{eHS} with
$\tau=2T$. Let $X=H_0^1([ 0,T ];\mathbb{R}^n)$ be a Hilbert space with the
inner product 
$( u,\omega  )=\int_0^{T}[ \dot{u}( t )\dot{\omega }( t )+u( t
)\omega ( t ) ]dt$ and the corresponding norm
\[
\| u \|_{\Delta }=( u,u )^{1/2}=( \int_0^{T}[ | \dot{u}( t ) |^2
+| u( t ) |^2 ]dt )^{1/2}. 
\]
The Pointcare inequality  
$\int_0^{T}| u( t ) |^2dt\leqslant ( \frac{T}{\pi} )^2\int_0^{T}| 
\dot{u}( t ) |^2dt$ implies that 
\[
\| u \|=(\int_0^{T}| \dot{u}( t ) |^2dt)^{1/2}
\]
 is also a norm in $X$, and is equivalent to $\| u \|_{\Delta }$. 
 Now we define the functional $\varphi _{T}( u )$ on $X$:
\begin{equation}
\varphi _{T}( u )
=\frac{1}{2}\int_0^{T}| \dot{u}( t ) |^2dx-\int_0^{T}V(t,u( t )  )dt
\equiv \frac{1}{2}\| u \|^2-J( u )\label{e19}
\end{equation}
From $V( t,u )\in \mathbb{C}^1( \mathbb{R}\times \mathbb{R}^n )$, we 
know that $\varphi _{T}\in \mathbb{C}^1( X,\mathbb{R})$, and ${J}'( u )$ 
is completely continuous.

\begin{proof}[Proof of Theorem \ref{thm2}]
 Under the assumptions of Theorem \ref{thm2},  
$\{ \sin\frac{j\pi t}{T}e \}_{j=1}^{\infty }$ is an orthogonal sequence with 
$e=(1,0,0,\dots,0)\in \mathbb{R}^n$ in both $X$ and $L^2$ such that
$\| \sin\frac{j\pi t}{T}e \|_{L^2( 0,T )}^2= \frac{T}{2}$,
$\| sin\frac{j\pi t}{T}e \|^2=\frac{T}{2}( \frac{j\pi }{T} )^2$, for all 
$j\geqslant 1$, the functional \eqref{e19} satisfies 
$( J_{1})-(J_{3} )$ of Theorem \ref{thm1}. 
Thus, $\varphi _{T}( u )$ possesses at least $p$ distinct pairs 
$( u_{i},-u_{i} )$ of critical points such that 
$\varphi _{T}( u_{i})< 0$ with $u_{i}\neq 0( 1\leqslant i\leqslant p )$. 
Since $X\cap \mathbb{R}^n=0$,we conclude that $u_{i}\neq$ any
constant $( 1\leqslant i\leqslant p )$. Thus, in the way
of \eqref{e18}, the extensions of $\pm u_{i}( t )$ $(1\leqslant i\leqslant p )$ 
are $p$ distinct pairs of nontrivial odd $\tau -$ periodic solutions of
\eqref{eHS}.
\end{proof}

\begin{remark}[\cite{l1}] \label{rmk4} \rm
 For  $\alpha \in ( 0,1/2)$, we can choose a function 
$h\in\mathbb{C}^1( [ 0,\infty ),\mathbb{R} )$
such that 
\begin{gather*} 
r^{1+2\alpha }\leqslant h( r)\leqslant r^{1+\alpha }\quad \text{for }
0\leqslant r \leqslant 1 ,\\
-r^{4}\leqslant h( r ) \leqslant \frac{1}{8}r^2\quad \text{for } r\geqslant 2.
\end{gather*}
Define $V(t,u )= ( 1+\frac{1}{2}\cos t )h( | u | )$ then for
$\tau =2\pi$, for all $p\geqslant 1$, (V1)-(V3) are satisfied.
Thus, by Theorem \ref{thm2}, \eqref{eHS} has
infinitely many nontrivial pairs $(u( t ),-u( t ) )$ of odd 
$2\pi$-periodic solutions.
\end{remark}

Finally, since the proof of Theorem \ref{thm3},
is similar to that of Theorem 2, so we briefly sketch it.

\begin{proof}[Proof of Theorem \ref{thm3}]
 Set
\begin{equation}
X=H^2( 0,T )\cap H_0^1( 0,T )\label{e20}
\end{equation}
and the functional
\begin{equation}
\varphi _{T}( u )= \frac{1}{2}\int_0^{T}|
\ddot{u}( t ) |^2dx-\int_0^{T}V(t,u( t ) )dt,u\in X.\label{e21}
\end{equation}
Then the critical points of $\varphi _{T}$ in \eqref{e21}
are the classical solutions of the problem \eqref{eFE}. 
By \cite[Lemma 2.1]{g1},
\[
\| u \|=( \int_0^{T}| \ddot{u}( t ) |^2dt )^{1/2}
\]
is a norm in $X$, 
$\| u \|_{L^2( 0,T )}\leqslant ( \frac{T}{\pi } )^2\| u \|$, and the
set of functions $\{ \sin\frac{j\pi t}{T} \}_{j=1}^{\infty }$ is  an 
orthogonal sequence in both $X$ and $L^2$  such that  
\[
\| \sin\frac{j\pi t}{T} \|_{L^2( 0,T )}^2= \frac{T}{2}, \quad
\|\sin\frac{j\pi t}{T} \|^2= \frac{T}{2}( \frac{j\pi}{T} )^{4},\quad
\forall j\geqslant 1.
\]
Therefore, by Theorem \ref{thm1}, the proof is complete.
\end{proof}

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous referees
for their valuable suggestions.

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