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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 120, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/120\hfil the Linking method ]
{Linking method for periodic non-autonomous fourth-order
differential equations with superquadratic potentials}

\author[C. Li, C. Shi \hfil EJDE-2009/120\hfilneg]
{Chengyue Li, Changhua Shi}  % in alphabetical order

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

\address{Changhua Shi \newline
Department of Mathematics, Minzu University of China, Beijing
100081, China}
\email{shichanghua88888@163.com}

\thanks{Submitted June 18, 2009. Published September 27, 2009.}
\thanks{Supported by SRF for ROCS, SEM (2007-2008),
and ``211 Engineer'' Project from \hfill\break\indent
the Ministry of Education in China}
\subjclass[2000]{58E05, 34C37, 70H05}
\keywords{Periodic solutions; fourth-order differential equations;
\hfill\break\indent linking theorem; critical points}

\begin{abstract}
 By means of the Schechter's Linking method, we study the
 existence of $2T$-periodic solutions of the non-autonomous
 fourth-order ordinary differential equation
 $$
 u''''-Au''-Bu-V_u(t,u)=0
 $$
 where $A>0$, $B>0$, $V(t,u)\in \mathbb{C}^1(\mathbb{R}\times\mathbb{R},
 \mathbb{R})$ is $2T$-periodic in $t$ and satisfies either
 $0<\theta V(t,u) \leq u V_u(t,u)$ with  $\theta>2$, or
 $u V_u(t,u)-2V(t,u)\geq d_3|u|^r$ with $r\geq1$.
\end{abstract}

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

\section{Introduction}\label{Sec:Intro}

Pulse propagation through optical fibers involving a fourth-order
negative dispersion term leads to a generalized nonlinear
Schrodinger equation \cite{a1,b3}. After an appropriate
scaling of the variables this equation takes the form
\begin{equation}\label{1}
i\frac{\partial w}{\partial x}+\frac{\partial^2 w}{\partial
t^2}-\frac{\partial^4 w}{\partial t^4}+|w|^2w=0.
\end{equation}
Considering harmonic spatial dependence $w(t,x)=u(t)e^{ikx}$
with $k<0$, one obtains
\begin{equation}\label{2}
u^{(4)}-u''+ku-u^3=0.
\end{equation}
Motivated by \eqref{2}, we shall discuss the more
general equation
\begin{equation}\label{3}
u^{(4)}-Au''-Bu-V_u(t,u)=0,
\end{equation}
where $A>0$, $B>0$, the potential
$V(t,u)\in\mathbb{C}^1(\mathbb{R}\times\mathbb{R},\mathbb{R})$,
$V_u(t,u)=\partial V(t,u)/\partial u$.

Indeed, many other types of fourth-order differential equation
models in physical, chemical or biological systems have been studied
for recent years. We give some examples as follows:

(i) The equation $u^{(4)}-\gamma u''-u+u^3=0$ serves
as a model in studies of pattern formation and phase transitions
near Lifshitz points.
 If $\gamma >0$, it is the Extended Fisher-Kolmogorov equation
 proposed by Dee  and Saarloos  van in \cite{d1}. If $\gamma <0$,
 it is the Swift-Hohenberg equation which has been proposed by
Swift  and Hohenberg  \cite{s4}.
 For the existence of its periodic solutions, we refer the readers to \cite{p1}.

(ii) In the theory of shallow water waves driven by
gravity and capillarity, the equation $u^{(4)}+p u''+u-u^2=0$ has
been studied with $p<0$ \cite{b1}, which was extensively
considered by Buffoni \cite{b2}.

 (iii) Chen  and McKenna \cite{c1} studied the equation
$u^{(4)}+c^2 u''+V'(u)=0$ under
the assumptions that $V\in \mathbb{C}^2(\mathbb{R})$ is a potential
such that $V'(u)=(u+1)_+-1+g(u)$ with $|g''(u)|\leq K$ for
some $K>0$ .This result was improved by Smets  and Van den Berg
\cite{s3} for almost every $c\in [-\sqrt{4\alpha},\sqrt{4\alpha}]$,
assuming that $\limsup_{u\to \infty}V(u)/|u|^2=0$.

(iv) Tersian and Chaparova \cite{t1} studied the
equation $u^{(4)}+pu''+a(x)u-b(x)u^2-c(x)u^3=0$
 where $a(x),b(x), c(x)$ are periodic, and
$0<a_1\leq a(x)$, $0<c_1\leq c(x)$.
 They obtained the existence of periodic solutions of the
equation for $p\neq0$.

(v) Gyulov and Tersian \cite{g1} discussed the
equation $u^{(4)}+au''+bu+V_u(t,u)=0$ where $V(t,u)\geq c|u|^p$
with $p>2$, and obtained the existence and nonexistence of nontrival
periodic solutions of the equation by Brezis-Nirenberg's linking
Theorem and minimizing methods.

In the present paper, we
shall study the existence of periodic solutions of the
non-autonomous fourth-order equation \eqref{3}. Our main results are
as follows:

\begin{theorem}\label{dl1}
Let $A>0$, $B>0$. Assume that
$V(t,u)\in C^1({\mathbb{R}} \times {\mathbb{R}},{\mathbb{R}})$
satisfies the assumptions:
\begin{itemize}
\item[(V1)]  $V(t,u)=V(t+2T,u),V(t,u)=V(t,-u)$, for all
$t\in {\mathbb{R}},u\in {\mathbb{R}}$;

\item[(V2)]  $V(t,u)=o(|u|^2)$, as $u\to 0$ uniformly in
$t \in {\mathbb{R}}$;

\item[(V3)] There exists a constant $\theta >2$ such that
$$
0<\theta V(t,u)\leq uV_u(t,u),\quad \forall
t\in {\mathbb{R}},u\in {\mathbb{R}}\setminus\{0\}.
$$
\end{itemize}
Then \eqref{3} has at least one nontrivial
$2T$-periodic solution, provided that $\frac{T}{T_1}\notin
\mathbb{N}$ with $T_1=\pi\sqrt{2}/\sqrt{-A+\sqrt{A^2+4B}}$.
\end{theorem}

\begin{theorem}\label{dl2}
Let $A>0$, $B>0$. Suppose that $V(t,u)\in C^1({\mathbb{R}} \times
{\mathbb{R}},{\mathbb{R}})$ satisfies that {\rm (V1), (V2)} and the
following conditions:
\begin{itemize}

\item[(V3')]  $V(t,u)/| u|^2\to\infty$, as
$|u|\to\infty$ uniformly in $t\in {\mathbb{R}}$;

\item[(V4)] There are constants $\mu,d_1,d_2>0$ such that
$|V_u(t,u)|\leq d_1|u| ^\mu+d_2$, for all
$t\in{\mathbb{R}},u\in {\mathbb{R}}$;

\item[(V5)] There are constants $h,d_3>0,r\geq max \{1,\mu\}$
such that
$$
uV_u(t,u)-2V(t,u)\geq d_3| u |^r,\quad
\forall t\in {\mathbb{R}},\; |u|>h.
$$
\end{itemize}
Then the conclusion of Theorem \ref{dl1} holds.
\end{theorem}

\begin{remark}\label{yl1} \rm
Hypothesis (V3) is so-called  Ambrosetti-Rabinowitz
superquadratic condition which implies that there exist constants
$r_1>0$, $r_2>0 $ such that
\begin{equation}\label{7}
V(t,u)\geq r_1| u|^\mu -r_2, \quad \forall t\in
{\mathbb{R}},\; u\in \mathbb{R}.
\end{equation}
By direct computation we notice that, for example,
$V(t,u)=u^2 \ln (1+u^{2i})\ln (1+2u^{2j})$ or
$V(t,u)=u^2 \ln (1+u^{2i})$ ($i,j\in \mathbb{N}$) satisfies
(V3'), (V4),  and (V5), but does
not satisfy \eqref{7}. Therefore,  Theorems \ref{dl1} and
\ref{dl2} study two types of superquadratic
nonlinearities.
\end{remark}

\section{Preliminaries}

To study the existence of $2T$-periodic solutions of \eqref{3},
we first consider the solvability of the two-point boundary problem
\begin{equation}\label{eq.p}
\begin{gathered}
u^{(4)}-Au''-Bu-V_u(t,u)=0,\quad 0<t<T;\\
u(0)=u(T)=0,u''(0)=u''(T)=0.
\end{gathered}
\end{equation}
We shall obtain $2T$-periodic solutions of \eqref{3} which are
antisymmetric with respect to $t=0$ and $t=T$ taking the
$2T$-periodic extension of the odd extension
\begin{equation}\label{8}
\bar{u}(t)=\begin{cases}
 u(t), & 0\leq t \leq T; \\
-u(-t),&-T \leq t\leq 0
\end{cases}
\end{equation}
of the solution $u(t)$ for problem \eqref{eq.p}.

Assume $X=H^2([0,T])\cap H^1_0([0,T])$ be a Hilbert space with the
inner product
\begin{equation}\label{9}
(u,v)=\int_0^T(u''(t)v''(t)+u'(t)v'(t)+u(t)v(t))dt,
\end{equation}
which corresponds the norm
\[
 \| u\|_X=\Big(\int_0^T(| u''(t)| ^2+|
u'(t)| ^2+| u(t)| ^2)dt\Big)^{1/2}.
\]
 From the Poincare inequality
\begin{equation}\label{10}
\int_0^T| u(t)| ^2dt\leq \frac{T^2}{\pi^2}\int_0^T| u'(t)|^2dt,\quad
\int_0^T| u(t)| ^2dt\leq \frac{T^4}{\pi^4}\int_0^T| u''(t)|^2dt,
\end{equation}
we know that $\|u\|_X$,
\begin{gather}\label{11}
 \| u\|=(\int_0^T(|u''(t)| ^2dt)^{1/2},\\
\label{12}  \| u\|_*=(\int_0^T(|
u''(t)| ^2+A| u'(t)| ^2)dt)^{1/2},
\end{gather}
are equivalent norms in $X(T)$. In addition, an important fact in
$X(T)$ is that the set of functions
$\{sin\frac{k\pi t}{T}\}_{k=1}^{\infty}$is a complete
orthogonal basis \cite{g1}.

A function $u\in X(T)$ is said to be a weak solution of
\eqref{eq.p}, if
\[
\int_0^T(u''(t)v''(t)+Au'(t)v'(t)-Bu(t)v(t))dt
- \int_0^T V_u(t,u)vdt=0,\quad \forall v \in X(T).
\]
Define the pertinent functional
\begin{equation} \label{13}
I(u;T)=\int_0^T\frac{1}{2}(u''^{2}+Au'^{2}-Bu^2)dt-\int_0^T
V(t,u)dt,\quad\forall u \in X(T).
\end{equation}
Under the assumption of $V(t,u)\in \mathbb{C}^1(\mathbb{R}
\times\mathbb{R},\mathbb{R})$, we easily show
that $I(u;T)\in C^1(X(T),R)$ and
\begin{equation}\label{14}
I' (u;T)v=\int_0^T(u''v''+Au'v'-Buv)dt
- \int_0^T V_u(t,u)vdt=0,\quad \forall u,v\in X(T).
\end{equation}
So weak solutions of \eqref{eq.p} are critical points of $I(u;T)$.
In fact, by the standard way, weak solutions of \eqref{eq.p} are
exactly its classical solutions.

For $u\in X(T)$, using Fourier series, we have
\begin{gather}\label{15}
u=\sum _{k=1}^{\infty}c_k \sin(\frac{k\pi t}{T}), \\
\label{16}
I(u;T)=\frac{T}{4}\sum_{k=1}^{\infty}c_k^2P_k(T)-\int_0^T V(t,u)dt,
\end{gather}
where $P_k(T)=P(\frac{k\pi}{T})$ with
$P(\xi)=\xi^4+A\xi^2-B,\,\xi\in \mathbb{R}$. Clearly for every
$T>0$,
\begin{equation}\label{17}
P_1(T)<P_2(T)<P_3(T)<\dots < P_n(T))<\dots.
\end{equation}
For every $n\in \mathbb{N}$, the equation $P_n(T)=0$ has the unique
solution
\begin{equation}\label{18}
T_n=nT_1,\quad T_1= \pi\sqrt{2}/ \sqrt{-A+\sqrt{A^2+4B}},
\end{equation}
and $P_n(T)>0$, if $T<nT_1$; $P_n(T)<0$, if $T>nT_1$.

To prove Theorems \ref{dl1} and \ref{dl2}, we shall use
linking method due to Schechter. For that, we start recalling the
definition of linking sets in the sense of homeomorphisms
\cite{s1}.

 Let $E$ be a real Banach space and let
$\Phi$ be the set of all continuous maps $\Gamma=\Gamma(t)$ from
$E\times[0,1]$ to $E$ such that
(i) $\Gamma(0)=I$, the identity map.
(ii) For each $t\in[0,1)$, $\Gamma (t)$ is a homeomorphism of $E$
into $E$ and $\Gamma^{-1}(t)\in \mathbb{C}(E\times[0,1],E)$.
(iii) $\Gamma(1)E$ is a single point in $E$ and $\Gamma(t)A$ converges
uniformly to $\Gamma(1)E$ as $t\to 1$ for each bounded set
$A\subset E$.
(iv) For each $t_0\in[0,1)$ and each bounded set
$Y\subset E$, $\sup_{0\leq t\leq t_0,u\in Y}\{\|\Gamma(t)u\|
+\|\Gamma^{-1}(t)u\|\}<\infty$.

We say that $Y$ links $Z$ if $Y$ and $Z$ are subsets of $E$ such
that $Y\cap Z=\phi$ and, for each $\Gamma\in\Phi$, there is a
$t\in(0,1]$ such that $\Gamma(t)Y\cap Z\neq \phi$.
Many examples of linking sets are presented in \cite{s1}.
A typical one is as follows:

\subsection*{Example} \cite[Example 3, P.38]{s1}.
Let $M$ and $N$ be closed subspaces of Banach space $E$ such that
$\dim N<\infty$ and $E=M\oplus N$. Let $w_0\neq0$ be an element of
$M$, $0<\rho<R$, and take
\begin{gather*}
Y=\{v\in N:\|v\|\leq R\}\cup\{v+\lambda
w_0:v\in N,\lambda\geq0,\|v+\lambda w_0\|=R\},\\
Z=\partial B_{\rho}(0)\cap M.
\end{gather*}
Then $Y$ links $Z$.

It was shown in \cite{s1} that with the aid of linking method a
deformation theorem was obtained and then, using standard minimax
arguments, the following result was proved by Schechter:



\begin{theorem}[Linking Theorem 2.1.1 and Corollary 2.8.2 in {[9]}]\label{dl3}
Assume that $E$ is a real Banach space, the functional $\varphi\in
\mathbb{C}^1(\mathbb{E},\mathbb{R})$. $Y$ and $Z$ are subsets of $E$
such that $Y$ is compact and $Y$ links $Z$, and satisfies that
$a_0:=sup_Y\varphi\leq b_0:=inf_Z\varphi$. If $a=inf_{\Gamma\in
\Phi}sup_{0\leq s\leq 1,u\in Y}\varphi(\Gamma(s)u)$ is finite, then
there is a sequence $(u_m)\subset E$ such that $\varphi(u_m)\to
a\geq b_0$, $(1+\|u_m\|)\varphi'(u_m)\to 0$. Furthermore, if
$a=b_0$, then $\mathop{\rm dist}(u_m,Z)\to 0$.
\end{theorem}

In addition, we also recall the limit case of Rabinowitz's Mountain
Pass Lemma, which shall be employed in the section 3 and section 4.

\begin{theorem}[\cite{r1}]\label{dl4}
 Let $E$ be a real Banach space
and $\varphi\in \mathbb{C}^1(\mathbb{E},\mathbb{R})$ satisfying the
$(PS)$ condition, $\varphi(0)=0$. If $\varphi$ satisfies
\begin{itemize}
\item[(a)]  There is an open neighborhood $Y$ of the origin $0$
such that $\varphi|_{\partial Y}\geq0$;

\item[(b)] There is $e\notin\overline{Y}$ such that
 $\varphi(e)\leq0$,
\end{itemize}
then $\varphi$ possesses a critical value $b\geq0$ at the level
characterized by
\[
b=\sup_{Z\in W}\inf_{u\in \partial Z}\varphi(u),
\]
 where
$W=\{Z\subset E:Z\text{ is open $0\in Z$ and $e\notin \overline{Z}$}\}$.
Moreover, if  $b=0$, there is a critical point of
$\varphi$ on $\partial Y$.
\end{theorem}

\section{Proof of Theorem \ref{dl1}}

\begin{lemma}\label{tl1}
 Under the assumptions of Theorem \ref{dl1}, the $(PS)$ condition
holds for $I(u;T)$. Namely, if $(u_m)\subset X(T)$ satisfies that
\begin{equation}\label{19}
|I(u_m;T)|\leq M_1,\quad |I'(u_m;T)|\to 0,
\end{equation}
for some constant $M_1>0$,then there is a subsequence of $(u_m)$
converging to a limit $u_0\in X(T)$.
\end{lemma}

\begin{proof}
 Choose $\theta^*\in (2,\theta)$.  By
(V3), \eqref{7} and \eqref{19}, we have
\begin{equation}\label{20}
\begin{aligned}
M_1+\|u_m\|
&\geq I(u_m;T)-\frac{1}{\theta^*}I'(u_m;T)u_m \\
&= \frac{1}{2}\int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt-\int_0^TV(t,u_m)dt \\
&\quad -\frac{1}{\theta^*}\Big(\int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt
 -\int_0^TV_u(t,u_m)u_mdt\Big)  \\
&= (\frac{1}{2}-\frac{1}{\theta^*})\int_0^T(u_m^{''2}+Au_m^{'2})dt
  -(\frac{1}{2}-\frac{1}{\theta^*})\int_0^TBu_m^2dt\\
&\quad +\int_0^T(\frac{V_u(t,u_m)u_m}{\theta^*}-V(t,u_m))dt \\
&\leq (\frac{1}{2}-\frac{1}{\theta^*})\int_0^T(u_m^{''2}+Au_m^{'2})dt
 -(\frac{1}{2}-\frac{1}{\theta^*})\int_0^TBu_m^2dt\\
 &\quad +\frac{\theta-\theta^*}{\theta^*}\int_0^TV(t,u_m)dt \\
&\leq (\frac{1}{2}-\frac{1}{\theta^*})\int_0^T(u_m^{''2}+Au_m^{'2})dt
 -(\frac{1}{2}-\frac{1}{\theta^*})\int_0^TBu_m^2dt\\
&\quad  +r_1\frac{\theta-\theta^*}{\theta^*}\|u_m\|^{\theta}_{L^\theta}
  -Tr_2\frac{\theta-\theta^*}{\theta^*} \\
&\leq (\frac{1}{2}-\frac{1}{\theta^*})\int_0^T(u_m^{''2}
  +Au_m^{'2})dt-(\frac{1}{2}-\frac{1}{\theta^*})B\|u_m\|^2_{L^2}\\
 &\quad +r_3\|u_m\|^{\theta}_{L^2}-Tr_2\frac{\theta-\theta^*}{\theta^*}
\end{aligned}
\end{equation}
with $r_3>0$. We claim that $\|u_m\|_{L^2}$ is bounded.
Otherwise, $\|u_m\|_{L^2}\to\infty$,
$\|u_m\|\to\infty$. Thus, since $\theta>2$, for $m$
sufficiently large, we have
\begin{equation}\label{21}
-(\frac{1}{2}-\frac{1}{\theta^*})B\|u_m\|_{L^2}^2
+r_3\|u_m\|_{L^2}^{\theta}-Tr_2\frac{\theta-\theta^*}{\theta^*}>0.
\end{equation}
Consequently, by \eqref{20} and \eqref{21}, we deduce that
\[
M_1+\|u_m\|\geq(\frac{1}{2}-\frac{1}{\theta^*})
\int_0^T(u_m^{''2}+Au_m^{'2})dt\geq(\frac{1}{2}
-\frac{1}{\theta^*})\|u_m\|^2,
\]
which contradicts $\|u_m\|\to\infty$. So $\|u_m\|_{L^2}$ is
bounded. Therefore, by \eqref{20}, there exists $M_2>0$ such that
\begin{equation}\label{22}
M_1+\|u_m\|\geq(\frac{1}{2}-\frac{1}{\theta^*})\int_0^Tu_m^{''2}dt+M_2
=(\frac{1}{2}-\frac{1}{\theta^*})\|u_m\|^2+M_2,
\end{equation}
This inequality implies $\|u_m\|$ is bounded in $X(T)$. Then we can
assume that, without loss of generation,
\begin{equation} \label{23}
u_m\rightharpoonup u_0 \in X(T),\quad
u_m\to u_0 \in \mathbb{C}([0,T]).
\end{equation}
So, by \eqref{14} and \eqref{23}, we have
\begin{equation}\label{24}
\begin{aligned}
&\|u_m-u_0\|^2+A\int_0^T|u_m'-u_0'|^2dt \\
&= B\int_0^T|u_m-u_0|^2dt+(I'(u_m)-I'(u_0))(u_m-u_0) \\
&\quad +\int_0^T(V_u(t,u_m)-V_u(t,u_0))(u_m-u_0)dt\to 0,
\end{aligned}
\end{equation}
namely, $u_m\to u_0$ in $X(T)$.
\end{proof}

\begin{lemma}\label{tl2}
Under the assumptions of Theorem \ref{dl1}, if $T>T_1$ and
$\frac{T}{T_1}\notin \mathbb{N}$, then the functional $I(u;T)$
possesses a nontrivial critical point in $X(T)$.
\end{lemma}

\begin{proof}
 There exists $n\in \mathbb{N}$ such that $nT_1<T<(n+1)T_1$. Define
\begin{gather}\label{25}
E_n=  span\{sin\frac{\pi t}{T},sin\frac{2\pi t}{T}\dots
sin\frac{n\pi t}{T}\},\\
 Y= \{v\in E_n:\|v\|\leq R\}\cup\{v+\lambda e:v\in
E_n,\lambda\geq0,\|v+\lambda e\|=R\},\\
Z= \partial B_\rho(0)\cap E_n^{\perp}.\label{27}
\end{gather}
where $e\in E_n^{\perp}$, $\|e\|=1$ and $0<\rho<R$.
By the typical example in section 2, $Y$ links with $Z$.
We shall verify for $R$ sufficiently large and $\rho$ sufficiently small,
that the following inequality holds:
\begin{equation}\label{28}
\sup_YT(u;T)\leq 0\leq\inf_ZI(u;T).
\end{equation}
Firstly, for every $v\in E_n$,
$v(t)=\sum_{k=1}^nc_k \sin(\frac{k\pi t}{T})$,
since
\begin{equation}\label{29}
P_1(T)<P_2(T)<P_3(T)<\dots P_n(T)<0,
\end{equation}
we have
\begin{equation}\label{30}
I(v;T)=\frac{T}{4}\sum_{k=1}^nP_k(T)c_k^2-\int_0^TV(t,v(t))dt\leq0.
\end{equation}
Secondly, we take $e=c_{n+1}\sin\frac{(n+1)\pi t}{T}\in E_n^{\perp}$
with $c_{n+1}$ such that $\|e\|=1$, and let
$$
w=u+\lambda e=\sum_{k=1}^nc_k
\sin(\frac{k\pi t}{T})+\lambda e,\lambda \geq0.
$$
Then $\|w\|=\|u\|+\|\lambda e\|=\|u\|+\lambda$.
There exist $r_4,r_4'$ such that
$r_4'\|w\|_{L^{\theta}}\leq \|w\|\leq r_4\|w\|_{L^{\theta}}$,
for all $w\in E_{n+1}$. Therefore, by
\eqref{7}, we conclude that
\begin{equation} \label{31}
\begin{aligned}
I(w;T)&= \int_0^T\frac{1}{2}((w^{''2}+Aw^{'2}-Bw^2))dt
 -\int_0^TV(t,w)dt \\
&= \frac{T}{4}\sum_{k=1}^nP_k(T)c_k^2
 +\frac{T}{4}P_{n+1}(T)\lambda^2c^2_{n+1}-\int_0^TV(t,w)dt \\
&\leq \frac{T}{4}P_{n+1}(T)\lambda^2c^2_{n+1}-\int_0^TV(t,w)dt \\
&\leq \frac{T}{4}P_{n+1}(T)c^2_{n+1}\|w\|^2-r_1r_4^{-\theta}
 \|w\|^{\theta}+r_2T\leq0
\end{aligned}
\end{equation}
for $\|w\|=R$ large enough. Finally, by
(V2) for each $\varepsilon>0$, there is $\delta\in (0,1)$ such
that
$$
|V(t,u)|\leq \varepsilon|u|^2\quad\text{if $|u|\leq\delta$ and
$t\in [0,T]$}.
$$
By the Sobolev embedding Theorem, there exists a constant
$r_5>0$ such that
\begin{equation}\label{32}
\|u\|_{C([0,T])}=\|u\|_{L^{\infty}[0,T]}\leq r_5\|u\|,\quad
\forall u\in X(T).
\end{equation}
 Let $0<\rho<\min\{\delta/r_5,R\}$ and
$\|u\|=\rho$, then $|u(t)|\leq \delta$ for all $t\in [0,T]$.
Therefore,
\[
\int_0^TV(t,u(t))dt\leq \varepsilon\|u\|^2_{L^2}.
\]
Noticing $0<p_{n+1}(T)<p_{n+2}(T)\dots$ for
$u\in E_n^{\perp}\cap B_{\rho}(0)$,
$u=\sum_{k=n+1}^{\infty}c_k\sin(\frac{k\pi t}{T})$,
we have
\begin{equation} \label{33}
\begin{aligned}
I(u;T)&= \frac{T}{4}\sum_{k=n+1}^{\infty}P_k(T)c_k^2
  -\int_0^TV(t,u(t))dt, \\
&\leq P_{n+1}(T)\|u\|^2_{L^2}-\varepsilon\int_0^T|u(t)|^2dt, \\
&\leq \frac{1}{2}P_{n+1}(T)\|u\|^2_{L^2}\geq0
\end{aligned}
\end{equation}
if $0<\varepsilon<\frac{1}{2}P_{n+1}(T)$.
Then \eqref{30}, \eqref{31} and \eqref{33} imply that \eqref{28} holds.
Thus, by Theorem \ref{dl3}, there exists a sequence $(u_m)\subset
X(T)$ satisfies that
\begin{gather}\label{34}
I(u_m;T)\to d_0\geq0, \\
\label{35}
(1+\|u_m\|)I'(u_m;T)\to 0.
\end{gather}
By Lemma \ref{tl1}, we may assume that
$u_m\to u_0\in X(T)$. And, by \eqref{35}, we can show that
$u_0$ is a critical point of $I(u;T)$. If $d_0>0$,
then $u_0\neq 0$. If $d_0=0$, then
$\mathop{\rm dist}(u_m,Z)\to 0$ by Theorem \ref{dl3}.
Hence there is a sequence $(v_m)\subset Z$ such that
$u_m-v_m\to 0$  in $X(T)$, so $v_m\to u_0$, thus
$\|u_0\|=\lim_{m\to\infty}\|v_m\|=\rho\neq0$.
\end{proof}

\begin{lemma}\label{tl3}
Under the assumptions of Theorem \ref{dl1}, if  $0<T<T_1$, then the
functional $I(u;T)$ possesses a nontrivial critical point in $X(T)$.
\end{lemma}

\begin{proof}
 We shall use Theorem \ref{dl4} to prove the existence of
the critical point of $I(u;T)$. Under the condition $0<T<T_1$, we have
\begin{equation}\label{36}
0<P_1(T)<P_2(T)<P_3(T)<\dots P_n(T)<\dots.
\end{equation}
Similar to Lemma \ref{tl2}, for
$0<\varepsilon<\frac{1}{2}P_1(T)$, there exists  $ \delta\in (0,1)$
such that $V(t,u)\leq\varepsilon|u|^2$
if $|u|\leq\delta$ and $t\in[0,T]$.
Then for every $u=\sum_{k=1}^{\infty}c_k\sin(\frac{k\pi t}{T})\in X(T)$
 such that $\|u\|=\rho <\delta/r_5$, where
$r_5$ is defined in \eqref{32}, we have
\begin{equation} \label{37}
\begin{aligned}
I(u;T)&= \frac{T}{4}\sum_{k=1}^{\infty}P_k(T)c_k^2
 -\int_0^T(V(t,u(t))dt, \\
&\leq P_1(T)\|u\|^2_{L^2}-\varepsilon\int_0^T|u(t)|^2dt, \\
&\leq \frac{1}{2}P_1(T)\|u\|^2_{L^2}
\end{aligned}
\end{equation}
which is non-negative.
Next, for some $\overline{u}\in E\backslash \{0\}$ and all $\sigma>0$,
we have
\begin{equation} \label{38}
\begin{aligned}
I(\sigma\overline{u};T)
&= \frac{\sigma^2}{2}\int_0^T(\overline{u}^{''2}+
A\overline{u}^{'2}-B\overline{u}^2)dt-\int_0^TV(t,\sigma\overline{u})dt \\
&\leq \frac{\sigma^2}{2}(\int_0^T(\overline{u}^{''2}+A\overline{u}^{'2}
 -B\overline{u}^2))dt-r_1\sigma^{\theta}
 \int_0^T|\overline{u}|^{\theta}dt+r_2T.
\end{aligned}
\end{equation}
Then $I(\sigma\overline{u};T)\to -\infty$ as
$\sigma\to \infty$. Hence, by Lemma \ref{tl1} and Theorem
\ref{dl4}, the functional $I(u;T)$ has at least one nontrivial
critical point in $X(T)$.
\end{proof}

The proof of Theorem \ref{dl1} follows from combining
Lemmas \ref{tl1}, \ref{tl2} and \ref{tl3}.

\section{Proof of Theorem \ref{dl1}} %1.2

\begin{lemma} \label{tl4}
Under the assumptions of Theorem \ref{dl2}, if $T>T_1$ and
 $\frac{T}{T_1}\notin\mathbb{N}$, then the functional $I(u;T)$
possesses a nontrivial critical point in $X(T)$.
\end{lemma}

\begin{proof}
 In the same way as \eqref{25}-\eqref{27}, we define
$E_n$,  $Y$ and $Z$. Under the assumptions of Theorem \ref{dl2},
we can verify that
 \eqref{30}, \eqref{31} and \eqref{33} still hold, whose proofs
are  similar to that of Lemma \ref{tl1} with
the exception of the inequality \eqref{31} resulting from (V3).
In its place we proceed as follows

Still take $e=c_{n+1}\sin\big(\frac{(n+1)\pi t}{T}\big)\in E_n^{\perp}$
with $c_{n+1}$ such that $\|e\|=1$, and let
$$
w=u+\lambda e=\sum_{k=1}^nc_k \sin(\frac{k\pi t}{T})
+\lambda e,\lambda \geq0.
$$
Then $\|w\|=\|u\|+\|\lambda e\|=\|u\|+\lambda$.
There exist $r_6,r_6'>0$ such that
$r_6'\|w\|_{L^2}\leq \|w\|\leq r_6\|w\|_{L^2}$, for all
$w\in E_{n+1}$.

By (V3'), there exists $r_7>0$ such that
\begin{equation}
V(t;u)\geq(\frac{T}{4}P_{n+1}(T)c_{n+1}^2r_6^2+1)|u|^2-r_7,\quad
\forall t\in \mathbb{R},u\in \mathbb{R}.
\end{equation}
Therefore,
%\label{40}
\begin{align*}
I(w;T)&= \frac{1}{2}\int_0^T(w^{''2}+Aw^{'2}-Bw^2))dt-\int_0^TV(t,w)dt \\
&= \frac{T}{4}\sum_{k=1}^nc_k^2P_k(T)
 +\frac{T}{4}\lambda^2P_{n+1}(T)c_{n+1}^2-\int_0^TV(t,w)dt \\
&\leq \frac{T}{4}\lambda^2P_{n+1}(T)c_{n+1}^2-\int_0^TV(t,w)dt \\
&\leq \frac{T}{4}P_{n+1}(T)c_{n+1}^2r_6^2\|w\|^2_{L^2}
  -(\frac{T}{4}P_{n+1}(T)c_{n+1}^2r_6^2+1)\|w\|^2_{L^2}+r_7T \\
&= -\|w\|^2_{L^2}+r_7T  \\
&\leq -r_6^{-2}\|w\|^2+r_7T\to -\infty\quad (\text{as }\|w\|\to \infty).
\end{align*}
Hence, under the assumptions of Theorem \ref{dl2}, $Y$ links $Z$,
so, by Theorem \ref{dl3}, there exists a sequence $(u_m)\subset
X(T)$ such that \eqref{34} and \eqref{35} hold. We shall prove that
$(u_m)$ is bounded in $X(T)$. If not, we may assume that
$\|u_m\|\to\infty$. From $(V_5)$, we get
\begin{equation} \label{41}
\begin{aligned}
2I(u_m;T)-I'(u_m;T)u_m
&= \int_0^T(u_m(t)V_u(t,u_m(t))-2V(t,u_m(t)))dt, \\
&\leq d_3\int_{|u_m(t)|\geq h}|u_m(t)|^rdt+d_4.
\end{aligned}
\end{equation}
with $d_4$ being a constant. By \eqref{34}, \eqref{35} and
\eqref{41}, we obtain
\begin{equation}\label{42}
\frac{1}{\|u_m\|} \int_{|u_m(t)|\geq h}|u_m(t)|^rdt\to 0.
\end{equation}
On the other hand, in view of (V4), we have
\begin{equation} \label{43}
\begin{aligned}
&I'(u_m;T)u_m \\
&= \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt-\int_0^Tu_m(t)V_u(t,u_m)dt \\
&\leq \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt-d_1\int_0^T|u_m(t)|^{\mu+1}dt-d_2\int_0^T|u_m(t)|dt \\
&= \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt-d_1(\int_{|u_m(t)|\geq h}|u_m(t)|^{\mu+1}dt \\
&\quad +\int_{|u_m(t)|\leq h}|u_m(t)|^{\mu+1}dt)-d_2\int_0^T|u_m(t)|dt \\
&\leq \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt-d_1\|u_m\|_{L^{\infty}}
 \int_{|u_m(t)|\geq h}|u_m(t)|^{\mu}dt\\
 &\quad -d_2\|u_m\|_{L^1}-d_5 \\
&\leq \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt
 -d_6\|u_m\|\int_{|u_m(t)|\geq h}|u_m(t)|^{\mu}dt\\
 &\quad -d_7\|u_m\|-d_5 \\
&\leq \int_0^T(u_m^{''2}+Au_m^{'2}-Bu_m^2)dt
-d_6h^{\mu-r}\|u_m\|\int_{|u_m(t)|\geq
h}|u_m(t)|^rdt\\
&\quad -d_7\|u_m\|-d_5,
\end{aligned}
\end{equation}
with $d_5,d_6,d_7$ being positive constants. The two sides of
\eqref{43} are divided by $\|u_m\|^2$, by \eqref{12}, we have
\begin{equation}\label{44}
\begin{aligned}
\frac{I'(u_m;T)u_m}{\|u_m\|^2}
&\geq \frac{\|u_m\|^2_{*}}{\|u_m\|^2}
 -B\int_0^T(\frac{u_m}{\|u_m\|})^2dt\\
&\quad -d_6h^{\mu-r}\frac{\int_{|u_m(t)|
\geq h}|u_m(t)|^rdt}{\|u_m\|}-\frac{d_7\|u_m\|+d_5}{\|u_m\|^2}.
\end{aligned}
\end{equation}
Set  $\widetilde{u_m}(t)=\frac{u_m(t)}{\|u_m\|}$, then
$\widetilde{u_m}(t)=1$. We may assume that
$\widetilde{u_m}(t)\rightharpoonup \chi\in
X(T)$ and $\widetilde{u_m}(t)\to \chi $ in
$\mathbb{C}([0,T])$, and
$\frac{\|u_m\|_{*}}{\|u_m\|}\to\tau>0$. Letting
$m\to\infty$ in \eqref{44}, and by \eqref{41}, we have
$B\int_0^T(\chi(t))^2dt\geq \tau^2>0$, which implies the measure of
$\Omega:= \{t\in [0,T]:\chi(t))\neq0\}$ is positive. For
every $t\in \Omega$, we have
$|u_m(t)|=\|u_m\||\widetilde{u_m}(t)|\to\infty$, so by
\eqref{34}, \eqref{35} and (V5), we have
\begin{equation} \label{45}
\begin{aligned}
2d_0&\leftarrow 2I(u_m;T)-I'(u_m;T)u_m \\
&= \int_0^T(u_m(t)V_u(t,u_m(t))-2V(t,u_m(t)))dt, \\
&= \int_{\Omega}(u_m(t)V_u(t,u_m(t))-2V(t,u_m(t)))dt\\
&\quad +\int_{[0,T]\backslash\Omega}(u_m(t)V_u(t,u_m(t))-2V(t,u_m(t)))dt, \\
&\leq d_3\int_{\Omega}|u_m(t)|^rdt+\text{a bounded term}
\to\infty,
\end{aligned}
\end{equation}
which is a contradiction. Therefore, $(u_m)$ is bounded in $X(T)$.
Referring to \eqref{23}-\eqref{24}, we can show that $u_m$ converges
to some critical point $u_0$ of $I(u;T)$ in $X(T)$. Following the
proof of Lemma \ref{tl2}, we also have $u_0\neq 0$.
\end{proof}

\begin{lemma}\label{tl5}
Under the assumptions of Theorem \ref{dl2}, if $0<T<T_1$, then the
functional $I(u;T)$ possesses a nontrivial critical point in $X(T)$.
\end{lemma}

 The proof of the above lemma is simple, so we omit it; see also
Lemma \ref{tl3}.

The Proof of Theorem \ref{dl2} follows from Lemma \ref{tl4} and
\eqref{tl5}.

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee
for the valuable suggestions.


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\end{document}