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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 205, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/205\hfil Homoclinic solutions]
{Homoclinic solutions for a class of second-order
Hamiltonian systems with locally defined potentials}

\author[X. Lv \hfil EJDE-2017/205\hfilneg]
{Xiang Lv}

\address{Xiang Lv \newline
Department of Mathematics,
Shanghai Normal University, Shanghai 200234,  China}
\email{lvxiang@shnu.edu.cn}

\thanks{Submitted  March 13, 2017. Published September 7, 2017.}
\subjclass[2010]{34C37, 70H05, 58E05}
\keywords{Homoclinic solutions; Hamiltonian systems; variational methods}

\begin{abstract}
 In this article, we  establish  sufficient conditions for the existence
 of homoclinic solutions for a class of second-order Hamiltonian systems
 $$
 \ddot u(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=f(t),
 $$
 where $L(t)$ is a positive definite symmetric matrix for all $t\in\mathbb{R}$.
 It is worth pointing out that the potential function $W(t,u)$ is locally
 defined and can be superquadratic or subquadratic with respect to $u$.
\end{abstract}

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

\section{Introduction and statement of main results}

The purpose of this article is to investigate the second-order 
Hamiltonian systems
\begin{equation}
\ddot u(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=f(t)
\label{HS}
\end{equation}
where $t\in\mathbb{R}$, $u\in\mathbb{R}^{n}$, 
$L\in C(\mathbb{R},\mathbb{R}^{n\times n})$ is a positive definite and
symmetric matrix for all $t\in\mathbb{R}$, 
$W:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ and 
$f:\mathbb{R}\to\mathbb{R}^{n}$. Here, we say that a solution $u(t)$ 
of \eqref{HS} is nontrivial homoclinic (to 0) if $u\not\equiv0$ and
$u(t)\to 0$ as $t\to \pm\infty$. Moreover, $\nabla W(t,x)$
denotes the gradient with respect to $x$, $(\cdot,\cdot):\mathbb{R}^n\times
\mathbb{R}^n\to\mathbb{R}$ denotes the standard inner product in $\mathbb{R}^n$ 
and $|\cdot|$ is the induced norm.

If $f=0$, then \eqref{HS} degenerates to the following second-order Hamiltonian 
system
\begin{equation}
\ddot u(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=0
\label{HS2}
\end{equation}

In physics, Hamiltonian systems describe the evolution equations of a physical 
system, which can present important insight about the dynamics, even if the 
analytical solution of the initial value problem cannot be obtained.
It is well known that a homoclinic orbit lies in the intersection of the 
stable manifold and the unstable manifold of a saddle point, which is a 
fundamental tool in the study of chaos. In the past decades, there have been 
a lot of results about the existence and multiplicity of homoclinic orbits 
for Hamiltonian systems via critical point theory, see 
\cite{AC,AR,CM, CES,CR,D,DG1,IJ,IJ2,KL,L,Lv,LJ,LLY,MW,OW,OT,R,R2,RT,SCN,T,W,ZL,ZY,Z}
and the references therein.

In the case that $L(t)$ and $W(t,x)$ are either independent
of $t$ or periodic in $t$, it has been studied by many authors, see
\cite{AC,CES,DG1,IJ,IJ2,L,OT,OW,R,T}. In particular, in \cite{R}, Rabinowitz has
proved the existence of homoclinic orbits as a limit of
$2kT$-periodic solutions of \eqref{HS2}. Motivated by the work of Rabinowitz, 
applying the same procedure, the existence of homoclinic solutions of \eqref{HS} 
or \eqref{HS2} was obtained as the limit of subharmonic solutions, see
Izydorek and Janczewska \cite{IJ,IJ2} and so on.

In the case that $L(t)$ and $W(t,x)$ are not periodic with respect to $t$, 
the problem of existence and multiplicity of
homoclinic orbits for \eqref{HS} will become much more difficult, due to 
the lack of compactness of the Sobolev
embedding. In \cite{RT}, Rabinowitz and Tanaka considered \eqref{HS2}
without a periodicity assumption, both for $L$ and $W$. 
To deal with the case that the nonlinearity $W$ is superquadratic, 
they introduced the Ambrosetti-Rabinowitz growth condition, i.e., 
the following assumption (A1) and
assumed that the smallest eigenvalue of $L(t)$ tends
to $+\infty$ as $|t|\to \infty$. Using a variant of the Mountain
Pass theorem without the Palais-Smale condition, they proved that
\eqref{HS2} possesses a nontrivial homoclinic
orbit.

For the next theorem we use  the following assumptions:

\begin{itemize}
\item[(A1)] $L(t)$ is positive definite symmetric matrix for all $t\in\mathbb{R}$
and there exists an $l\in C(\mathbb{R},\bigl(0,\infty)\bigr)$ such that
$l(t)\to +\infty$ as $|t|\to \infty$ and
\[
\bigl(L(t)x,x\bigr)\geq l(t)|x|^{2}\quad\text{for all $t\in\mathbb{R}$
 and $x\in\mathbb{R}^{n}$};
\]

\item[(A2)]
 $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and there is a constant 
$\mu>2$ such that
      \[
0<\mu W(t,x)\leq\bigl(x,\nabla W(t,x)\bigr)\quad\text{for all $t\in\mathbb{R}$
 and $x\in\mathbb{R}^{n}\setminus\{0\}$};
\]

\item[(A3)]
 $|\nabla W(t,x)|=o(|x|)$ as $|x|\to 0$ uniformly with respect to $t\in\mathbb{R}$;

\item[(A4)] There is a $\overline W\in C(\mathbb{R}^{n},\mathbb{R})$ such that
     \[
|W(t,x)|+|\nabla W(t,x)|\leq|\overline W(x)|\quad \text{for all 
$t\in\mathbb{R}$  and $x\in\mathbb{R}^{n}$}.
\]
\end{itemize}

\begin{theorem}[\cite{RT}] \label{thm1}
Assume that $L$ and $W$ satisfy {\rm (A1)--(A4)}.
Then \eqref{HS2} possesses a nontrivial homoclinic solution.
\end{theorem}

Motivated by \cite{L,RT}, in this paper, we  study the existence
of Homoclinic solutions for \eqref{HS}, where we only give some local 
assumptions on $W(t,u)$ and $W(t,u)$ can be superquadratic or subquadratic 
with respect to $u$. Our main results are stated in the next  theorem,
under the following conditions:

\begin{itemize}
\item[(A5)] $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$,
 $W(t,0)\equiv0$ and $\nabla W(t,0)\equiv0$ for all $t\in\mathbb{R}$;

\item[(A6)] there exist $\rho>0$ and $a\in
L^\alpha(\mathbb{R},\mathbb{R}_+)$ such that
\begin{equation}\label{eq0}
W(t,x)\leq a(t)|x|^\mu\quad\text{for all $t\in\mathbb{R}$ and $|x|\leq\rho$},
\end{equation}
where $\alpha>1$, $\mu>1$ if $\frac{2(\alpha-1)}{\alpha}\leq1$ or 
$\mu\geq\frac{2(\alpha-1)}{\alpha}$ if $\frac{2(\alpha-1)}{\alpha}>1$.

\item[(A7)] $f\not\equiv0$ is a continuous and bounded function such
that $\int_\mathbb{R}|f(t)|^\beta dt<\infty$ and
\begin{equation}\label{ineq1}
\frac{1\wedge l_*}{4}\rho-\frac{M_a}{\sqrt[\alpha^*]{2}}\rho^{\mu-1}
-\frac{M_f}{\sqrt[\beta^*]{2}}>0,
\end{equation}
where $1<\beta\leq2$, $\frac{1}{\alpha^*}+\frac1\alpha=1$, 
$\frac{1}{\beta^*}+\frac1\beta=1$, $l_*=\inf_{t\in\mathbb{R}}l(t)>0$,
\[
M_a=\Big(\int_\mathbb{R}|a(t)|^\alpha dt\Big)^{1/\alpha}\quad\text{and}
\quad M_f=\Big(\int_\mathbb{R}|f(t)|^\beta dt\Big)^{1/\beta}.
\]
\end{itemize}

\begin{theorem}\label{thm2}
Assume that  {\rm (A1), (A5)--(A7)}.
Then \eqref{HS} possesses a nontrivial homoclinic
solution.
\end{theorem}

\section{Proof of main results}

Motivated by \cite{KL,LJ}, we first consider the existence of the homoclinic
 solutions for \eqref{HS}, which can be obtained as the limit of periodic 
solutions for the following boundary-value problem
\begin{equation}\label{eq1}
\begin{gathered}
\ddot u(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=f(t),\quad t\in [-T,T]\\
u(-T)-u(T)=\dot u(-T)-\dot u(T)=0,
\end{gathered}
\end{equation}
for all $T\in\mathbb{R}^+$.

Given any $T\in\mathbb{R}^+$, let
\begin{align*}
E_{T}:=&W^{1,2}\bigl([-T,T],\mathbb{R}^{n}\bigr)\\
=& \big\{u: [-T,T]\to \mathbb{R}^{n}: u\text{ is absolutely continuous},\\
&\; u(-T)=u(T)\text{ and }
\dot u\in L^{2}([-T,T],\mathbb{R}^{n})\big\}
\end{align*}
and for $u\in E_{T}$, define
\[
\|u\|_{E_T}=\Big\{\int^{T}_{-T}[|\dot
u(t)|^{2}+|u(t)|^2]dt\Big\}^{1/2},
\]
then $E_{T}$ is a Hilbert space endowed with the above norm.

Next, we define a functional $I_T : E_{T}\to\mathbb{R}$ by
\begin{equation}\label{eq2}
I_T(u)=\int_{-T}^{T}\big[\frac12|\dot u(t)|^2+\frac12\bigl(L(t)u(t),u(t)\bigr)
-W\bigl(t,u(t)\bigr)+\bigl(f(t),u(t)\bigr)\big]dt.
\end{equation}
We can easily see that $I_T\in C^{1}(E_{T},\mathbb{R})$ is weakly lower
semi-continuous because it is the sum of a convex continuous function 
and of a weakly continuous one. By the direct calculation, it follows that
\begin{equation}\label{eq3} 
\begin{aligned}
\langle I_{T}'(u),v\rangle 
&=\int^{T}_{-T}\big[\big(\dot u(t),\dot v(t)\big)+\big(L(t)u(t),v(t)\big)\\
&\quad -\bigl(\nabla W\bigl(t,u(t)\bigr),v(t)\bigr)+\bigl(f(t),v(t)\bigr)\big]dt
\end{aligned}
\end{equation}
for all $u,v\in E_{T}$. Moreover, it is well known that the critical
points of $I_T$ in $E_{T}$ are classical solutions of \eqref{eq1} (see
\cite{MW,R2}).

To prove our main result, we  apply a critical point theorem,
 which is stated precisely as follows.

\begin{lemma}[ See \cite{L}]\label{lem1}
Let $X$ be a real reflexive Banach
space and $\Omega\subset X$ be a closed bounded convex subset of $X$.
Suppose that $\varphi: X\to\mathbb{R}$ is weakly lower
semi-continuous. If there exists a point
$x_0\in\Omega\setminus\partial\Omega$ such that
\begin{equation}\label{eq4}
\varphi(x)>\varphi(x_0)\quad{\rm for\ all}\ x\in\partial\Omega.
\end{equation}
Then there exists a $x^\ast\in\Omega\setminus\partial\Omega$ such
that
\[
\varphi(x^\ast)=\inf_{u\in\Omega}\varphi(u).
\]
\end{lemma}

\begin{lemma}[See \cite{IJ}]\label{lem2}
Let $u:\mathbb{R}\to\mathbb{R}^n$ be a continuous
mapping such that $\dot u\in L_{loc}^2(\mathbb{R},\mathbb{R}^n)$. 
Then for every $t\in \mathbb{R}$, we have
\begin{equation}\label{eq5}
|u(t)|\leq\sqrt2\Big[\int_{t-\frac12}^{t+\frac12}\big(|\dot
u(s)|^{2}+|u(s)|^{2}\big)ds\Big]^{1/2}.
\end{equation}
\end{lemma}

\begin{lemma}\label{lem3}
Let $u\in E_T$. It follows that
\begin{equation}\label{eq6}
\|u\|_{L_{[-T,T]}^\infty}
\leq\Big(\int_{-T}^{T}|u(t)|^2dt\Big)^{1/2}
+\Big(\int_{-T}^{T}|\dot u(t)|^2dt\Big)^{1/2}.
\end{equation}
\end{lemma}

Note that the above lemma is a special case of \cite[Corollary 2.2]{T}.

\begin{corollary}\label{cor1} 
Let $u\in E_T$. It follows that
\begin{equation}\label{eq7}
\|u\|_{L_{[-T,T]}^\infty}\leq\sqrt2\|u\|_{E_T}
=\sqrt2\Big\{\int^{T}_{-T}\big[|\dot
u(t)|^{2}+|u(t)|^2\big]dt\Big\}^{1/2}.
\end{equation}
\end{corollary}

\begin{proof}
Combining \eqref{eq6} and the inequality $\sqrt a+\sqrt b\leq\sqrt2(a+b)^{1/2}$,
it is obvious that \eqref{eq7} holds.
\end{proof}

\begin{lemma}\label{lem4}
Under the conditions of Theorem \ref{thm2},  the boundary-value problem \eqref{eq1} 
admits a solution $u_T\in E_T$ such that
\begin{equation}\label{eq8}
\int_{-T}^{T}\left[|\dot
u_T(t)|^{2}+|u_T(t)|^2\right]dt<\frac12\rho^2\quad {\rm for\ all}\
T\in\mathbb{R}_+.
\end{equation}
\end{lemma}

\begin{proof} 
Clearly, $I_T(0)=0$ by  (A5) for all $T\in\mathbb{R}_+$. For the purpose 
of using Lemma \ref{lem1}, we first need to construct
a closed bounded convex subset of $E_T$ for all $T\in\mathbb{R}_+$. 
Given any $T\in\mathbb{R}_+$,
let $\Omega_T:=\{u\in E_T : \int_{-T}^{T}\left[|\dot
u(t)|^2dt+|u(t)|^2\right]dt\leq\frac12\rho^2\}$, where $\rho$ is the
constant defined in \eqref{eq0}. It is evident that $\Omega_T$ is a
closed bounded convex subset of $E_T$ for all $T\in\mathbb{R}_+$.

For any $T\in\mathbb{R}_+$, we will prove that \eqref{eq8} holds. 
If $u\in\partial\Omega_T$, it follows that
$\int_{T}^{T}\left[|\dot u(t)|^2dt+|u(t)|^2\right]dt=\frac12\rho^2$.
Applying Corollary \ref{cor1}, it is obvious that 
$\| u\|_{L_{[-T,T]}^{\infty}}\leq\rho$ for all
$u\in\partial\Omega_T$. That
is $|u(t)|\leq\rho$ for all $t\in [-T,T]$. Combining this inequality,
 (A1), (A6) and (A7), we get that $\mu\geq\frac{2}{\alpha^*}$ and
\begin{align*}
&I_T(u) \\
&=\int_{-T}^{T}\Big[\frac12|\dot
u(t)|^2+\frac12\bigl(L(t)u(t),u(t)\bigr)-W\bigl(t,u(t)\bigr)+\bigl(f(t),u(t)\bigr)
\Big]dt\\
&\geq\frac{1}{2}\int_{-T}^{T}|\dot
u(t)|^{2}dt+\frac{1}{2}\int_{-T}^{T}l(t)|u(t)|^2dt
-\int_{-T}^{T}a(t)|u(t)|^\mu dt+\int_{-T}^{T}\bigl(f(t),u(t)\bigr)dt\\
&\geq\frac{1}{2}\int_{-T}^{T}|\dot
u(t)|^{2}dt+\frac{l_*}{2}\int_{-T}^{T}|u(t)|^2dt
 -\Big(\int_{-T}^{T}|a(t)|^\alpha dt\Big)^{1/\alpha}
 \big(\int_{-T}^{T}|u(t)|^{\mu\alpha^*} dt\Big)^{1/\alpha^*}\\
&\quad-\Big(\int_{-T}^{T}|f(t)|^\beta dt\Big)^{1/\beta}
 \Big(\int_{-T}^{T}|u(t)|^{\beta^*} dt\Big)^{1/\beta^*}\\
&\geq\frac{1}{2}\int_{-T}^{T}|\dot
u(t)|^{2}dt+\frac{l_*}{2}\int_{-T}^{T}|u(t)|^2dt
 -\|u\|_{L_{[-T,T]}^{\infty}}^{\mu-\frac{2}{\alpha^*}}
\Big(\int_\mathbb{R}|a(t)|^\alpha dt\Big)^{1/\alpha} \\
&\quad\times \Big(\int_{-T}^{T}|u(t)|^{2} dt\Big)^{1/\alpha^*}
 -\|u\|_{L_{[-T,T]}^{\infty}}^{1-\frac{2}{\beta^*}}
 \Big(\int_\mathbb{R}|f(t)|^\beta dt\Big)^{1/\beta}
  \Big(\int_{-T}^{T}|u(t)|^{2} dt\Big)^{1/\beta^*}\\
&\geq \frac{1\wedge l_*}{4}\rho^2-\frac{M_a}{\sqrt[\alpha^*]{2}}\rho^\mu
 -\frac{M_f}{\sqrt[\beta^*]{2}}\rho\\
&>0=I_T(0)
\end{align*}
for all $u\in\partial\Omega_T$. Consequently, using Lemma \ref{lem1}, 
we can have that
for all $T\in\mathbb{R}_+$, there exists $u_T\in\operatorname{int}\Omega_T$
such that
\[
I_T(u_T)=\inf_{u\in\Omega_T}I_T(u),
\]
where 
\[
\operatorname{int}\Omega_T=\Big\{u\in E_T: \int_{-T}^{T}[|\dot
u(t)|^2+|u(t)|^2]dt<\frac12\rho^2\Big\}
\]
Furthermore, we note that $\operatorname{int}\Omega_T$ is an open subset of
$E_T$. This together with \cite[Theorem 1.3]{MW} implies that
\[
I'_T(u_T)=0.
\]
That is, $u_T$ is the solution of the boundary-value problem \eqref{eq1} and
\[
\int_{-T}^{T}[|\dot u_T(t)|^2+|u_T(t)|^2]dt<\frac12\rho^2.
\]
The proof is complete.
\end{proof}

\subsection*{Proof of Theorem \ref{thm2}} 
First, we can choose a sequence $T_{m}\to\infty$ and
study the boundary-value problem \eqref{eq1} on the bounded closed interval 
$[-T_{m},T_{m}]$ for all $m\in\mathbb{N}$.
Using the result of Lemma \ref{lem4}, it follows that there exists a sequence 
of solutions $u_{m}$ such that
$\|u_{m}\|_{E_{T_m}}$ is uniformly bounded with respect to $m\in\mathbb{N}$.

According to the  inequality
\[
|u_{m}(t_{1})-u_{m}(t_{2})|\leq \int_{t_{1}}^{t_{2}}|\dot u_{m}(t)|dt
\leq\sqrt{t_{2}-t_{1}}\Big(\int_{t_{1}}^{t_{2}}|\dot
u_{m}(t)|^{2}dt\Big)^{1/2}
\]
we can assert that the sequence $\{u_{m}\}_{m\in\mathbb{N}}$ is equicontinuous 
and uniformly bounded on every bounded closed interval $[-T_{m},T_{m}]$, 
$m\in\mathbb{N}$. Therefore, we can select a subsequence
$\{u_{m_{k}}\}_{k\in\mathbb{N}}$ such that it converges uniformly on any 
bounded closed interval to a continuous function $u$. Furthermore, 
using \eqref{eq1}, it is clear that the sequence 
$\{\ddot u_{m_{k}}\}_{k\in\mathbb{N}}$
and so $\{\dot u_{m_{k}}\}_{k\in\mathbb{N}}$ converges uniformly on any 
bounded closed intervals.
Noting that
\[
u_{m_{k}}(t)=\int_{0}^{t}(t-s)\ddot
{u}_{m_{k}}(s)ds+t\dot{u}_{m_k}(0)+u_{m_k}(0),
\]
it is obvious that $u\in C^{2}(\mathbb{R},\mathbb{R}^{n})$ and 
$\ddot {u}_{m_{k}}\to\ddot {u}$ uniformly on any
bounded closed intervals as $k\to\infty$. Consequently, we can first study
the boundary-value problem \eqref{eq1} on bounded closed interval 
$[-T_{m},T_{m}]$, $m\in\mathbb{N}$.
Next, using the diagonal process and let $m\to\infty$, we can easily see 
that $u$ is a classical solution of \eqref{HS}.

Since $\|u_{m}\|_{E_{T_m}}$ is uniformly bounded with respect to $m\in\mathbb{N}$, 
under the above analysis, it is evident that
\begin{equation}\label{eq9}
\int_{\mathbb{R}}[|\dot u(t)|^{2}+|u(t)|^2]dt\leq\frac12\rho^2.
\end{equation}
By Lemma \ref{lem2}, we have
\[
|u(t)|\leq\sqrt2\Big[\int_{t-\frac12}^{t+\frac12}\big(|\dot
u(s)|^{2}+|u(s)|^{2}\big)ds\Big]^{1/2}
\quad \text{for all } t\in\mathbb{R}.
\]
This together with \eqref{eq9} implies that the limit of $u(t)$ is zero 
as $|t|\to \infty$, i.e., $u(\pm\infty)=0$.
 Moreover, since $f\not\equiv0$, it follows that
$u$ is a nontrivial homoclinic orbit of \eqref{HS}.

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation
of China (NSFC) under Grants No. 11371252 and No. 11501369, by the
Research and Innovation Project of Shanghai Education Committee under
 Grant No. 14zz120, by the  Yangfan Program of Shanghai (14YF1409100),
by the  Chen Guang Project(14CG43) of Shanghai Municipal Education Commission
and the  Shanghai Education Development Foundation,  by the Research Program
 of Shanghai Normal University (SK201403), and by the
Shanghai Gaofeng Project for University Academic Program Development.



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Manuscripta Math., 104 (2001), 343-358.

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