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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 93, pp. 1--15.\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-2017/93\hfil 
 Nonperiodic perturbed fractional Hamiltonian systems]
{Multiplicity of solutions for nonperiodic perturbed fractional
 Hamiltonian systems}

\author[A. Benhassine \hfil EJDE-2017/93\hfilneg]
{Abderrazek Benhassine}

\address{Abderrazek Benhassine \newline
Dept. of Mathematics,
High Institut of Informatics and Mathematics,
5000, Monastir, Tunisia}
\email{ab.hassine@yahoo.com}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted December 13, 2016. Published March 30, 2017.}
\subjclass[2010]{34C37, 35A15, 37J45}
\keywords{Fractional Hamiltonian systems; critical point; variational methods}

\begin{abstract}
 In this article, we prove the existence and multiplicity of nontrivial
 solutions for the nonperiodic perturbed fractional Hamiltonian systems
 \begin{gather*}
-_{t}D^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}x(t))
 -\lambda L(t)\cdot x(t)+\nabla W(t,x(t))=f(t),\\
 x\in H^{\alpha}(\mathbb{R}, \mathbb{R}^N),
 \end{gather*}
 where $\alpha \in (1/2 , 1]$, $\lambda> 0 $ is a parameter,
 $t\in \mathbb{R}, x\in \mathbb{R}^N$, ${}_{-\infty}D^{\alpha}_{t}$ and
 ${}_{t}D^{\alpha}_{\infty}$ are left and right Liouville-Weyl fractional
 derivatives of order $\alpha$ on the whole axis $\mathbb{R}$ respectively,
 the matrix $L(t)$ is not necessary positive definite for all $t\in \mathbb{R}$
 nor coercive, $W \in C^{1}(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$ and
 $f\in L^{2}(\mathbb{R},\mathbb{R}^N)\backslash\{0\}$ small enough.
 Replacing the Ambrosetti-Rabinowitz Condition by general superquadratic 
 assumptions,  we establish the existence and multiplicity results for the 
 above system.  Some examples are also given to illustrate our results.
\end{abstract}

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

\section{Introduction}

Hamiltonian systems form a significant field of nonlinear functional analysis,
since they arise in phenomena studied in
several fields of applied science such as physics, astronomy, chemistry,
biology, engineering and other fields of science. Since
Newton wrote the differential equation describing the motion of the planet
and derived the Kepler ellipse as its solution, the
complex dynamical behavior of the Hamiltonian system has attracted a wide
range of mathematicians and physicists. The
variational methods to investigate Hamiltonian system were first used
by Poincar\'e, who used the minimal action principle of
the Jacobi form to study the closed orbits of a conservative system with two
degrees of freedom. Ambrosetti and Rabinowitz
in \cite{AR} proved ``Mountain Pass Theorem'', ``Saddle Point Theorem'',
``Linking Theorem'' and a series of very important minimax
form of critical point theorem. The study of Hamiltonian systems makes a
significant breakthrough, due to critical point
theory. Critical point theorem was first used by Rabinowitz \cite{R} to
obtain the existence of periodic solutions for first order
Hamiltonian systems, while the first multiplicity result is due to Ambrosetti
and Zelati \cite{AZ}. Since then, there is a large number
of literatures on the use of critical point theory and variational methods
to prove the existence of homoclinic or heteroclinic
orbits of Hamiltonian systems see for example \cite{D, IJ, OW} and the
references therein.

On the other hand, fractional calculus has received increased popularity and
importance in the past decades to describe
long-memory processes. For more details, we refer the reader to the
 monographs \cite{ER, KST, MR} and the
reference therein. Recently, the critical point theory has become an
effective tool in studying the existence of solutions to
fractional differential equations by constructing fractional variational structures.

Recently, in Jiao and Zhou \cite{JZ} showed that critical point
theory is an effective approach to tackle the existence of solutions
for the fractional boundary-value problem
\begin{gather*}
 _tD^{\alpha}_{T}(_0D^{\alpha}_{t}x(t))=\nabla W(t,x(t)), \quad
\text{a.e. } t\in [0,T],\\
 x(0)=x(T),
 \end{gather*}
where $\alpha\in (1/2,1)$, $x\in \mathbb{R}^N$,
$W\in C^1([0,T]\times\mathbb{R}^N,\mathbb{R}), \nabla W(t,x)$
is the gradient of $W$ at $x$, and obtained the existence of at least one
nontrivial solution. Inspired by this paper, Torres \cite{T}
studied the fractional Hamiltonian system
\begin{equation}
\begin{gathered}
-_{t}D^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}x(t))-L(t)\cdot x(t)+\nabla W(t,x(t))=0,
\\
x\in H^{\alpha}(\mathbb{R}, \mathbb{R}^N),
\end{gathered} \label{FHS}
\end{equation}
where $\alpha \in (\frac12 , 1)$, $t\in \mathbb{R}, x\in \mathbb{R}^N$,
$_{-\infty}D^{\alpha}_{t}$ and $_{t}D^{\alpha}_{\infty}$ are left and
right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole
axis $\mathbb{R}$ respectively, $L(t)\in C(\mathbb{R},\mathbb{R}^{N^{2}})$
is symmetric and positive definite matrix for all $t\in\mathbb{R}$ and
$W \in C^{1}(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$.
The author showed that \eqref{FHS} possesses at least one nontrivial
solution via Mountain Pass Theorem, by assuming that $L$ and $W$ satisfy
the following hypotheses:
\begin{itemize}
\item[(A1)] $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a positive definite
 symmetric matrix for all $t\in \mathbb{R}$;

\item[(A2)] the smallest eigenvalue of $L(t)\to +\infty$ as $t\to \infty$;

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

\item[(A4)] there is $\overline{W} \in C(\mathbb{R}^N,\mathbb{R})$ such that
$$
|W(t,x)| +|\nabla W(t,x)| \leq |\overline{W}(x)|,\quad
 \forall (t,x) \in \mathbb{R}\times\mathbb{R}^N.
$$

\item[(A5)] there exists a constant $\mu>2$ such that
$$
 0<\mu W(t,x)\leq\nabla W(t,x)\cdot x,\quad \forall t\in \mathbb{R},\;
 x\in \mathbb{R}^N\backslash\{0\}.
 $$
\end{itemize}

When $\alpha=1$, \eqref{FHS} reduces to the standard second-order
Hamiltonian systems
\begin{equation}
\ddot{x}(t)- L(t)x(t)+\nabla W(t,x(t))=0. \label{HS}
\end{equation}
When $L(t)$ is a symmetric matrix valued function for all $t\in \mathbb{R}$
and $W(t,x)$ satisfies the so-called global Ambrosetti-Rabinowitz Condition (A5),
the existence and multiplicity of homoclinic solutions for Hamiltonian
systems \eqref{HS} have been extensively investigated in many recent papers
see for example \cite{AZ, D, IJ, RT} and the references therein.
If $L(t)$ and $W(t,x)$ are neither periodic in $t$, the problem of existence
 of homoclinic orbits for \eqref{HS} is quite different from the ones just
 described, because of lack of compactness of Sobolev embedding.
In \cite{RT} and without periodicity assumptions on both $L$ and $W$,
Rabinowitz and Tanaka first studied system \eqref{HS} and prove the existence
of one nontrivial homoclinic orbit of \eqref{HS} under assumptions
 (A1)--(A5).

\begin{remark} \label{rmk1.1} \rm
Although the technical coercively assumption (A2) plays a key role to guarantee
the compactness of the Sobolev embedding, it is somewhat restrictive and
eliminates many functions.
\end{remark}

Motivated by the above works , in this article, when
$f\in L^{2}(\mathbb{R},\mathbb{R}^N)\backslash\{0\}$,
$L(t) \in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix but not
 necessary positive definite for all $t\in \mathbb{R}$ not coercive,
 $W \in C^{1}(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$ and replacing
the Ambrosetti-Rabinowitz condition by general superquadratic assumptions,
 we establish the existence and multiplicity results for the
 nonperiodic perturbed fractional Hamiltonian system
\begin{equation}
\begin{gathered}
-_{t}D^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}x(t))-\lambda L(t)\cdot x(t)
+\nabla W(t,x(t))=f(t), \\
x\in H^{\alpha}(\mathbb{R}, \mathbb{R}^N),
\end{gathered} \label{PFHSlambda}
\end{equation}
where $\alpha \in (1/2 , 1], \lambda>0 $ is a parameter. Precisely,
we suppose that
\begin{itemize}
\item[(A6)] $\min_{x\in \mathbb{R}^N,|x|=1}L(t)x\cdot x\geq 0$ and there is $b>0$
such that
$\operatorname{meas}(\{ L\not\geq b\})< 1/c^{2}_{\alpha} $,
where $\operatorname{meas}(\cdot)$ is the Lebesgue measure,
$\{ L\not\geq b\}=\{t\in \mathbb{R}: L(t)\not\geq b\}$ and $c_{\alpha}$ defined
the Sobolev constant (see section 2);

\item[(A7)] $W(t,0)=0$ and for any $0<\alpha_1<\alpha_{2}$,
$$
C^{\alpha_1}_{\alpha_{2}}:=\inf\Big\{{\widetilde{W}(t,x)\over{|x|^{2}}};
t \in \mathbb{R},\alpha_1<|x|<\alpha_{2}\Big\}>0,
$$
 where $\widetilde{W}(t,x):= \frac12 \nabla W(t,x)\cdot x-W(t,x)$;

\item[(A8)] there exist $c_1>0$, $R_1>1$ and $\beta \in (1,2)$ such that
$$
\nabla W(t,x)\cdot x\leq c_1 \widetilde{W}(t,x)|x|^{2-\beta},\quad
 \forall t\in \mathbb{R}, |x|\geq R_1;
$$

\item[(A9)] there exist a constants $T_0>0$ and
$x_0\in \mathbb{R}^N\backslash\{0\}$ such that
$$
\int^{T_0}_{-T_0} \lambda L(t)x_0\cdot x_0 - W(t,x_0)\ dt <0.
$$
\end{itemize}
Our main results reads as follows.

\begin{theorem} \label{thm1.2}
 Assume that $f\in L^{2}(\mathbb{R},\mathbb{R}^N)\backslash\{0\}$ and
{\rm (A3), (A4), (A6)--(A9)} hold.
 Then, there exist constants $f_0, \lambda_0>0$ such that, for any
$ \lambda >\lambda_0$ system \eqref{PFHSlambda} possesses at least
two nontrivial solutions whenever $\|f\|_{L^{2}}<f_0$.
\end{theorem}

\begin{corollary} \label{coro1.3}
Assume that $f\in L^{2}(\mathbb{R},\mathbb{R}^N)\backslash\{0\}$,
{\rm(A3), (W2), (A6)--(A8)} are satisfied and
\begin{itemize}
\item[(A9')]
$$
\lim_{|x|\to +\infty}{W(t,x)\over{|x|^{2}}}=+\infty,\quad\text{uniformly for a.e. }
t \in \mathbb{R}.
$$
\end{itemize}
Then, there exist constants $f_0, \lambda_0>0$ such that, for any
$\lambda >\lambda_0$ system \eqref{PFHSlambda} possesses at least two
 nontrivial solutions whenever $\|f\|_{L^{2}}<f_0$.
\end{corollary}

\begin{remark} \label{rmk1.4} \rm
Assumption (A5) implies (A8), (A9) and (A9').
In fact assuming (A5) is satisfied, it is clear that (A9) and (A9') hold.
Choose $R_1\geq 1$ so large that
$$
\frac{1}{\mu}< \frac{1}{2}-\frac{1}{|x|^{2-\beta}} \quad\text{whenever }
 |x|\geq R_1.
$$
Then, for such $|x|$, we have
$$
W(t,x)\leq \Big(\frac{1}{2}-\frac{1}{|x|^{2-\beta}}\Big)\nabla W(t,x)\cdot x,
$$
and it follows that
$$
\nabla W(t,x)\cdot x \leq |x|^{2-\beta}\Big(\frac{1}{2}\nabla W(t,x)\cdot x- W(t,x)\Big)
=|x|^{2-\beta}\widetilde{W}(t,x).
$$
\end{remark}

Here and in the following $x\cdot y$ denotes the inner product of
$x,y \in \mathbb{R}^N$ and $|\cdot|$ denotes the associated norm.
Throughout this article, we denote by $c,c_i$ the various positive
constants which may vary from line to line and are not essential to the problem.

\section{Preliminaries}

\subsection{Liouville-Weyl Fractional Calculus}

\begin{definition}\label{def2.1} \rm
The left and right Liouville-Weyl fractional integrals of order $0<\alpha < 1$
on the whole axis $\mathbb{R}$ are defined by
\begin{gather*}
_{-\infty}I_{x}^{\alpha}u(x):= \frac{1}{\Gamma (\alpha)}
 \int_{-\infty}^{x}(x-\xi)^{\alpha - 1}u(\xi)d\xi,\\
_{x}I_{\infty}^{\alpha}u(x) := \frac{1}{\Gamma (\alpha)}
 \int_{x}^{\infty}(\xi - x)^{\alpha - 1}u(\xi)d\xi\,,
\end{gather*}
respectively, where $x\in \mathbb{R}$.
\end{definition}

\begin{definition} \label{def2.2} \rm
The left and right Liouville-Weyl fractional derivatives of order $0<\alpha < 1$
 on the whole axis $\mathbb{R}$ are defined by
\begin{gather}\label{eq2.1}
_{-\infty}D_{x}^{\alpha}u(x) := \frac{d }{d x} {_{-\infty}}I_{x}^{1-\alpha}u(x),\\
\label{eq2.2}
_{x}D_{\infty}^{\alpha}u(x) := -\frac{d }{d x} {_{x}}I_{\infty}^{1-\alpha}u(x)\,,
\end{gather}
respectively, where $x\in \mathbb{R}$.
\end{definition}

\begin{remark} \label{rmk2.3} \rm
Definitions \eqref{eq2.1} and \eqref{eq2.2} may be written in the
alternative forms:
\begin{gather*}
_{-\infty}D_{x}^{\alpha}u(x) = \frac{\alpha}{\Gamma (1-\alpha)}
\int_0^{\infty}\frac{u(x) - u(x-\xi)}{\xi^{\alpha + 1}}d\xi, \\
_{x}D_{\infty}^{\alpha}u(x) = \frac{\alpha}{\Gamma (1-\alpha)}
 \int_0^{\infty}\frac{u(x) - u(x+\xi)}{\xi^{\alpha + 1}}d\xi\,.
\end{gather*}
\end{remark}

Recall that the Fourier transform $\widehat{u}(w)$ of $u(x)$ is defined by
$$
\widehat{u}(w) = \int_{-\infty}^{\infty} e^{-ix.w}u(x)dx.
$$
We establish the Fourier transform properties of the fractional integral
and fractional operators as follows:
\begin{gather*}
\widehat{ _{-\infty}I_{x}^{\alpha}u(x)}(w) := (iw)^{-\alpha}\widehat{u}(w), \\
\widehat{ _{x}I_{\infty}^{\alpha}u(x)}(w) := (-iw)^{-\alpha}\widehat{u}(w), \\
\widehat{ _{-\infty}D_{x}^{\alpha}u(x)}(w) := (iw)^{\alpha}\widehat{u}(w), \\
\widehat{ _{x}D_{\infty}^{\alpha}u(x)}(w) := (-iw)^{\alpha}\widehat{u}(w).
\end{gather*}

\subsection{Fractional derivative spaces}

Let us recall for any $\alpha>0$, the semi-norm
$$
|u|_{I_{-\infty}^{\alpha}} := \|_{-\infty}D_{x}^{\alpha}u\|_{L^2},
$$
and the norm
\begin{equation*}
\|u\|_{I_{-\infty}^{\alpha}}
:= \Big( \|u\|_{L^2}^2 + |u|_{I_{-\infty}^{\alpha}}^2 \Big)^{1/2}\,.
\end{equation*}
Let the space $I_{-\infty}^{\alpha} (\mathbb{R})$ denote the completion
of $C^{\infty}_0(\mathbb{R})$ with respect to the norm $\|\cdot\|_{I_{-\infty}^{\alpha}}$,
i.e.,
$$
I_{-\infty}^{\alpha} (\mathbb{R})
= \overline{C_0^{\infty}(\mathbb{R})}^{\|\cdot\|_{I_{-\infty}^{\alpha}}}.
$$
Next, we define the fractional Sobolev space $H^{\alpha}(\mathbb{R})$ in terms
of the Fourier transform. For $0<\alpha<1$, define the semi-norm
\[
|u|_{\alpha} = \||w|^{\alpha}\widehat{u}\|_{L^2},
\]
and the norm
$$
\|u\|_{\alpha} = ( \|u\|_{L^2}^2 + |u|_{\alpha}^2 )^{1/2},
$$
and let
$$
H^{\alpha}(\mathbb{R}) := \overline{C_0^{\infty}(\mathbb{R})}^{\|\cdot\|_{\alpha}}.
$$
We note that a function $u\in L^{2}(\mathbb{R})$ belongs to
$I_{-\infty}^{\alpha} (\mathbb{R})$ if and only if
$$
|w|^{\alpha}\widehat{u} \in L^{2}(\mathbb{R}).
$$
In particular, $ |u|_{I_{-\infty}^{\alpha}}= \||w|^{\alpha}\widehat{u}\|_{L^{2}(\mathbb{R})}$.
Therefore $ H^{\alpha}(\mathbb{R})$ and $I_{-\infty}^{\alpha} (\mathbb{R})$ are
equivalent, with equivalent semi-norm and norm (see \cite{T}).

Analogous to $I_{-\infty}^{\alpha}(\mathbb{R})$, we introduce $I_{\infty}^{\alpha}(\mathbb{R})$.
Let us define the semi-norm
$$
|u|_{I_{\infty}^{\alpha}}:= \|_{x}D^{\alpha}_{\infty}\|_{ L^{2}(\mathbb{R})},
$$
and norm
$$
\|u\|_{I_{\infty}^{\alpha}}:=(\|u\|^{2}_{L^{2}}+|u|^{2}_{I_{\infty}^{\alpha}})^{1/2},
$$
and let
$$
I_{-\infty}^{\alpha} (\mathbb{R})
=\overline{C_0^{\infty}(\mathbb{R})}^{\|\cdot\|_{I^{\alpha}_{-\infty}}}.
$$
Moreover $I_{\infty}^{\alpha} (\mathbb{R})$ and $I_{-\infty}^{\alpha} (\mathbb{R})$
are equivalent, with equivalent semi-norm and norm.

\begin{lemma}[\cite{T}]  \label{lem2.4}
If $\alpha > 1/2$, then
$H^{\alpha}(\mathbb{R}) \subset C(\mathbb{R})$
and there is a constant $C=C_{\alpha}$ such that
\begin{equation}\label{eq2.3}
\|u\|_{L^{\infty}}=\sup_{x\in \mathbb{R}} |u(x)| \leq C \|u\|_{\alpha}
\end{equation}
where $C(\mathbb{R})$ denote the space of continuous functions on $\mathbb{R}$.
\end{lemma}

\begin{remark}\label{FDEnta01} \rm
If $u\in H^{\alpha}(\mathbb{R})$, then $u\in L^{q}(\mathbb{R})$ for
all $q\in [2,\infty]$, since
$$
\int_{\mathbb{R}} |u(x)|^{q}dx \leq \|u\|_{L^{\infty}}^{q-2}\|u\|_{L^2}^2\,.
$$
\end{remark}

In what follows, we introduce the fractional space in which we will construct
the variational framework of \eqref{PFHSlambda}. Let
$$
X^{\alpha} = \big\{ x\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{n}):
\int_{\mathbb{R}} |_{-\infty}D_{t}^{\alpha}x(t)|^2 + L(t)x(t)\cdot x(t) dt < \infty
 \big\}\,.
$$
The space $X^{\alpha}$ is a reflexive and separable Hilbert space with the
inner product
$$
(x,y)_{X^{\alpha}}
= \int_{\mathbb{R}} (_{-\infty}D_{t}^{\alpha}x(t). _{-\infty}D_{t}^{\alpha}y(t)) + L(t)x(t)\cdot y(t)dt,
$$
and the corresponding norm is
$$
\|x\|_{X^{\alpha}} = \sqrt{( x,x)_{X^{\alpha}}}.
$$
For $\lambda> 0$, we also need the following inner product
$$
(x,y)_{\lambda} = \int_{\mathbb{R}} (_{-\infty}D_{t}^{\alpha}x(t)\cdot{} _{-\infty}D_{t}^{\alpha}y(t) + \lambda L(t)x(t)\cdot y(t))dt,
$$
and the corresponding norm
$$
\|x\|_{\lambda} = \sqrt{( x,x)_{\lambda}}.
$$
Set ${X_{\lambda}^{\alpha}}=(X^{\alpha}, \|\cdot\|_{\lambda} )$. Observing
$\|x\|_{\lambda} \geq \|x\|_{X^{\alpha}} $ for all $\lambda\geq 1$.

\begin{lemma} \label{lem2.6}
 If $L$ satisfies {\rm (A6)} then, $X^{\alpha}$ is continuously embedded
in $H^{\alpha}(\mathbb{R},\mathbb{R}^{n})$.
\end{lemma}

\begin{proof}
By (A6) and \eqref{eq2.3} we have
\begin{align*}
&\int_{\mathbb{R}}|x(t)|^{2}dt \\
&=\int_{\{ L< b\}}|x(t)|^{2}dt + \int_{ \{ L\geq b\}}|x(t)|^{2}dt\\
&\leq \|x\|^{2}_{L^{\infty}} \operatorname{meas}(\{ L< b\})
+\frac{1}{b}\int_{ \{ L\geq b\}}L(t)x(t)\cdot x(t)dt \\
&\leq c_{\alpha}^{2} \operatorname{meas}(\{ L< b\})
\Big(\int_{\mathbb{R}}(|_{-\infty}D^{\alpha}_{t}x(t)|^{2}+|x(t)|^{2})dt\Big)
+\frac{1}{b}\int_{ \{ L\geq b\}}L(t)x(t)\cdot x(t)dt.
\end{align*}
 Therefore,
 \begin{equation}\label{eq2.4}
 \|x\|^{2}_{L^{2}}\leq {{\max \{ c_{\alpha}^{2} \operatorname{meas}(\{ L< b\}),
\frac{1}{b}\}}\over{1-c^{2}_{\alpha} \operatorname{meas}(\{ L< b\})} }
\|x\|^{2}_{X^{\alpha}}
 \end{equation}
 and
\begin{equation}\label{eq2.5}
\begin{aligned}
 \|x\|^{2}_{\alpha}
&= \int_{\mathbb{R}}(|_{-\infty}D^{\alpha}_{t}x(t)|^{2}+|x(t)|^{2})dt \\
&\leq \Big( 1+{{\max \{ c_{\alpha}^{2} \operatorname{meas}(\{ L< b\}),
\frac{1}{b}\}}\over{1-c^{2}_{\alpha} \operatorname{meas}(\{ L< b\})} }\Big)
\|x\|^{2}_{X^{\alpha}},
\end{aligned}
\end{equation}
which yields that the embedding
$X^{\alpha}\hookrightarrow H^{\alpha}(\mathbb{R},\mathbb{R}^N)$ is continuous.
\end{proof}

\begin{remark} \label{rmk2.7} \rm
 Using the same conditions and techniques in \eqref{eq2.4} and \eqref{eq2.5},
for all $\lambda \geq \frac{1}{bc^{2}_{\alpha}
 \operatorname{meas}(\{ L< b\})}$, we also obtain
\begin{gather}\label{eq2.6}
 \|x\|^{2}_{L^{2}}\leq \frac{c_{\alpha}^{2}
\operatorname{meas}(\{ L< b\})}{1-c^{2}_{\alpha} \operatorname{meas}(\{ L< b\})}
 \|x\|^{2}_{\lambda}, \\
\label{eq2.7}
 \|x\|^{2}_{\alpha} \leq \Big( 1+\frac{c_{\alpha}^{2}
\operatorname{meas}(\{ L< b\})}{{1-c^{2}_{\alpha} \operatorname{meas}
(\{ L< b\})} }\Big)
 \|x\|^{2}_{\lambda} .
\end{gather}
Furthermore, using \eqref{eq2.3}, \eqref{eq2.5} and \eqref{eq2.6}, for every
$p \in (2,\infty)$ and
\[
\lambda \geq \frac{1}{bc^{2}_{\alpha} \operatorname{meas}(\{ L< b\})},
\]
 we have
\begin{equation}\label{eq2.8}
\begin{aligned}
&\int_{\mathbb{R}}|x(t)|^{p}dt \\
&\leq \|x\|^{p-2}_{L^{\infty}}\int_{\mathbb{R}}|x(t)|^{2}dt \\
&\leq c^{p-2}_{\alpha}\Big(\int_{\mathbb{R}}(|_{-\infty}D^{\alpha}_{t}x(t)|^{2}
+|x(t)|^{2})dt\Big)^{\frac{p-2}{2}}
 \frac{c_{\alpha}^{2} \operatorname{meas}(\{ L< b\})}{1-c^{2}_{\alpha}
\operatorname{meas}(\{ L< b\})} \|x\|^{2}_{\lambda}\\
&\leq c^{p-2}_{\alpha}\Big( 1+\frac{c_{\alpha}^{2}
\operatorname{meas}(\{ L< b\})}{{1-c^{2}_{\alpha} \operatorname{meas}
(\{ L< b\})} }\Big)^{\frac{p-2}{2}} \|x\|^{p-2}_{\lambda}
\frac{c_{\alpha}^{2} \operatorname{meas}(\{ L< b\})}{1-c^{2}_{\alpha}
\operatorname{meas}(\{ L< b\})} \|x\|^{2}_{\lambda}
\\
& = \Big( 1+\frac{c_{\alpha}^{2} \operatorname{meas}(\{ L< b\})}{{1-c^{2}_{\alpha}
\operatorname{meas}(\{ L< b\})} }\Big)^{\frac{p-2}{2}}
 \frac{c_{\alpha}^{p} \operatorname{meas}(\{ L< b\})}{1-c^{2}_{\alpha}
\operatorname{meas}(\{ L< b\})} \|x\|^{p}_{\lambda} \\
& = \operatorname{meas}(\{ L< b\})
(\frac{c_{\alpha}^{2}}{1- c_{\alpha}^{2}\operatorname{meas}(\{ L< b\})})^{p/2}
 \|x\|^{p}_{\lambda}\\
& := \delta^{p}_{p} \|x\|^{p}_{\lambda}.
 \end{aligned}
\end{equation}
\end{remark}

\section{Proof of Theorem \ref{thm1.2} and Corollary \ref{coro1.3}}

For this purpose, we establish the corresponding variational framework to obtain
solutions of \eqref{PFHSlambda}. To this end,
define the functional $I_{\lambda}: X_{\lambda}^{\alpha} \to \mathbb{R}$ by
\begin{align*}
I_{\lambda}(x)
&= \int_{\mathbb{R}} \Big[ \frac{1}{2}|_{-\infty}D_{t}^{\alpha}x(t)|^2
 +\frac{\lambda}{2} L(t)x(t)\cdot x(t) - W(t,x(t))+f(t)\cdot x(t)\Big]dt \\
&= \frac{1}{2} \|x\|_{\lambda} ^2 - \int_{\mathbb{R}} W(t,x(t))dt
+ \int_{\mathbb{R}}f(t)\cdot x(t)dt.
\end{align*}
Under assumptions (A3), (A4), (A6)--(A8), we see that $I_{\lambda}$ is a
continuously Fr\'echet- differentiable functional
defined on $X_{\lambda}^{\alpha}$; i.e.,
$I_{\lambda}\in C^{1}(X_{\lambda}^{\alpha}, \mathbb{R})$. Moreover, we have
\begin{equation}
\begin{aligned}
&I_{\lambda}'(x)y \\
&= \int_{\mathbb{R}}
[ (_{-\infty}D_{t}^{\alpha}x(t). _{-\infty}D_{t}^{\alpha}y(t))
+\lambda L(t)x(t)\cdot y(t) - \nabla W(t,x(t))\cdot y(t) + f(t)\cdot y(t)]dt,
\end{aligned}
\end{equation}
for all $x,y \in X_{\lambda}^{\alpha}$, which yields
\begin{equation}
I_{ \lambda}'(x)x = \|x\|_{\lambda} ^2 - \int_{\mathbb{R}}\nabla W(t,x(t))\cdot
x(t)dt+ \int_{\mathbb{R}}f(t)\cdot x(t)dt.
\end{equation}
We know that to find a solutions of \eqref{PFHSlambda}, it suffices
to obtain the critical points of $I_{\lambda}$; see \cite{T}.
For this purpose the lemma below is useful.

Recall that $\phi \in C^{1}(E,\mathbb{R})$ satisfy the Palais-Smale
condition $(PS)$ if any sequence $(x_{n})\subset E$, for which
$(\phi(x_{n}))$ is bounded and $\phi'(x_{n})\to 0$ as $n\to \infty$,
possesses a convergent subsequence in $E$.

\begin{lemma}[\cite{R}] \label{lem2.8}
 Let $E$ be a real Banach space and $\phi \in C^{1}(E,\mathbb{R})$ satisfying
the Palais-Smale condition. If $\phi$ satisfies the following conditions:
\begin{itemize}
\item[(i)] $\phi(0)=0$,
\item[(ii)] there exist constants $\rho,\gamma>0$ such that
$\phi_{/\partial{B_{\rho}}(0)}\geq \gamma$,
\item[(iii)] there exist $e\in E\backslash {\overline{B}_{\rho}(0)}$
such that $\phi(e)\leq 0$.
\end{itemize}
Then $\phi$ possesses a critical value $c\geq \gamma$ given by
$$
c=\inf_{g\in \Gamma}\max_{s \in [0,1]}\phi(g(s)),
$$
where
$$
\Gamma=\{ g\in C([0,1],E): g(0)=0,g(1)=e \}.
$$
\end{lemma}

To find the critical points of $I_\lambda$, we shall show that
$I_\lambda$ satisfies the $(PS)$ condition.

Because of the lack of the compactness of the Sobolev embedding, we
require the following convergence result.

\begin{lemma} \label{lem2.3}
 Assume that $x_n\rightharpoonup x$ in $ X^{\alpha}_\lambda$,
{\rm (A3), (A4), (A7)} are satisfied and $f\in L^2$. Then
 \begin{gather}\label{eq3.3}
 I_\lambda(x_n-x)= I_{\lambda}(x_n)- I_\lambda(x)+o(1)\quad \text{as } n\to +\infty,\\
 \label{eq3.4}
 I'_\lambda(x_n-x)= I'_\lambda(x_n)- I'_\lambda(x)+o(1) \quad \text{as }
 n\to +\infty.
 \end{gather}
In particular, if $(x_n)$ is a $(PS)$ sequence of $ I_\lambda$ such that
$ I_\lambda(x_{n})\to c$ for some $c\in \mathbb{R}$ then
 \begin{gather}\label{eq3.5}
 I_\lambda(x_n-x)\to c-I_\lambda(x) \quad \text{as } n\to +\infty, \\
 \label{eq3.6}
 I'_\lambda(x_n-x)\to 0 \quad \text{as } n\to +\infty,
 \end{gather}
after passing to a subsequence.
 \end{lemma}

\begin{proof}
As $x_n\rightharpoonup x$ in $X^{\alpha}_\lambda$, we have
 $(x_{n},x)_{\lambda}\to (x,x)_{\lambda}$
 as $n\to \infty$. Then
 \begin{align*}
 \|x_{n}\|^{2}_{\lambda} 
&= (x_{n}-x,x_{n}-x)_{\lambda}+ (x,x_{n})_{\lambda}+(x_{n}-x,x)_{\lambda}\\
&= \|x_{n}-x\|^{2}_{\lambda} +\|x\|^{2}_{\lambda} +o(1).
\end{align*}
Obviously, 
$$ 
(x_{n},z)_{\lambda} =(x_{n}-x,z)_{\lambda}+(x,z)_{\lambda}, \quad \forall 
z \in X_{\lambda}^{\alpha}. 
$$
 Hence, to show \eqref{eq3.3} and \eqref{eq3.4} it suffices to prove that
 \begin{gather}\label{eq3.7}
 \int_{\mathbb{R}}( W(t,x_n)- W(t,x_n-x)- W(t,x)) dt=o(1),\\
\label{eq3.8}
 \sup_{\varphi \in X^{\alpha}_{\lambda},\|\varphi\|_{\lambda}=1 } 
\int_{\mathbb{R}}(\nabla W(t,x_n)-\nabla W(t,x_n-x)-\nabla W(t,x))
 \cdot \varphi dt=o(1).
 \end{gather}
Here, we only prove \eqref{eq3.8} the proof of \eqref{eq3.7} is similar.
Setting $y_n:=x_n-x$, then $y_n\rightharpoonup 0$ in $X^{\alpha}_\lambda$ 
and $y_n(t)\to 0$ a.e. $t\in\mathbb{R}$.
 From (A3), for every $\varepsilon>0$, there exist 
$\sigma=\sigma(\varepsilon) \in(0,1)$ such that
 \begin{equation}\label{eq3.9}
 |\nabla W(t,u)|\leq \varepsilon |u|, \quad \forall t\in\mathbb{R}, |u|\leq \sigma.
 \end{equation}
 By $(A4)$ and \eqref{eq3.9}, we have
 \begin{equation}\label{eq3.10}
 |\nabla W(t,u)|\leq \varepsilon |u|+c_\varepsilon |u|^2, \quad 
\forall t\in\mathbb{R}, |u|\leq N_1,
 \end{equation}
where 
\[
N_1:=\sup_n\{\|y_n\|_{L^{\infty}},\; \|y_n+x\|_{L^{\infty}},\; \|x\|_{L^{\infty}}+1\},\quad
c_\varepsilon=\max_{|u|\in[\sigma,N_1]}\overline{W}(u)\sigma^{-2}. 
\]
By \eqref{eq3.10} and the Young Inequality, for each 
$\varphi \in X^{\alpha}_\lambda$
 with $\|\varphi\|_{\lambda}=1$, we have
 \begin{align*}
&|(\nabla W(t,y_n+x)-\nabla W(t,y_n)).\varphi | \\
&\leq \varepsilon (|y_n+x|+|y_n|)|\varphi|+c_\varepsilon(|y_n+x|^2+|y_n|^2)|\varphi|,
 \\
&\leq c (\varepsilon |y_n||\varphi|+\varepsilon|x||\varphi|
 +c_\varepsilon |y_n|^2|\varphi|+c_\varepsilon |x|^2|\varphi| ),
 \\
&\leq c (\varepsilon |y_n|^2+\varepsilon|x|^2+\varepsilon |\varphi|^2
 +\varepsilon |y_n|^3+c'_\varepsilon|\varphi|^3+c''_\varepsilon|x|^3 ),
 \end{align*}
 and
 \begin{equation}\label{eq3.11}
\begin{aligned}
&|(\nabla W(t,y_n+x)-\nabla W(t,y_n)-\nabla W(t,x))\cdot \varphi | \\
&\leq c (\varepsilon |y_n|^2+\varepsilon |x|^2
 +\varepsilon |\varphi|^2+\varepsilon |y_n|^3
 +c'_\varepsilon|\varphi|^3+c''_\varepsilon|x|^3).
\end{aligned}
 \end{equation}
 If we take
 $$
\psi_n(t):= \max\{ |(\nabla W(t,y_n+x)-\nabla W(t,y_n)-\nabla W(t,x)) \varphi |
-c \varepsilon (|y_n|^2+|y_n|^3), 0\},
$$
 we obtain
 $$
0\leq \psi_n(t) \leq c (\varepsilon |x|^2
+\varepsilon |\varphi|^2+c'_\varepsilon|\varphi|^3
+c''_\varepsilon|x|^3)\in L^{1} (\mathbb{R},\mathbb{R}^N).
$$
 The Dominated Convergence Theorem implies that
 \begin{equation}\label{eq3.12}
 \int_{\mathbb{R}} \psi_n(t)\,dt\to 0\,\text{as}\, n\to \infty.
 \end{equation}
 It follows from the definition of $\psi_n(t)$ that
 $$ 
|(\nabla W(t,y_n+x)-\nabla W(t,y_n)-\nabla W(t,x)) .\varphi |
\leq \psi_n(t)+\varepsilon c (|y_n|^2+|y_n|^3),
$$
 and then
\begin{align*}
&|\int_{\mathbb{R}} (\nabla W(t,y_n+x)-\nabla W(t,y_n)-\nabla W(t,x))\cdot\varphi\,
 dt| \\
&\leq \| \psi_n(t)\|_{L^{1}}+\varepsilon c (\|y_n\|_{L^{2}}^2+\|y_n\|_{L^{3}}^3),
\end{align*}
 for all $n$. Because $\varphi$ is arbitrary in $X^{\alpha}_{\lambda}$, we obtain
\begin{align*}
&\sup_{\varphi\in X^{\alpha}_\lambda,\, \|\varphi\|_{\lambda}=1} 
\big|\int_{\mathbb{R}} (\nabla W(t,y_n+x)-\nabla W(t,y_n)
-\nabla W(t,x)) \cdot\varphi\,\, dt\big| \\
&\leq \| \psi_n(t)\|_{L^{1}}+\varepsilon c (\|y_n\|_{L{2}}^2+\|y_n\|_{L^{3}}^3),
\end{align*}
 which, jointly with \eqref{eq2.8} and \eqref{eq3.12} shows that
$$
\sup_{\varphi \in X^{\alpha}_\lambda,\, \|\varphi\|_{\lambda}=1} 
\big|\int_{\mathbb{R}} (\nabla W(t,y_n+x)-\nabla W(t,y_n)
-\nabla W(t,x))\cdot\varphi\, dt\big|
\leq \varepsilon c,
$$
 for $n$ sufficiently large. Therefore, \eqref{eq3.8} holds.

If moreover $I_\lambda(x_n)\to c$ and $I'_\lambda(x_n)\to 0$ as $n\to \infty$, 
equations \eqref{eq3.3} and \eqref{eq3.4} respectively, imply that
\[
 I_\lambda(x_n-x)\to c-I_\lambda(x)+o(1),
\]
 and
\[
 I'_\lambda(x_n-x)=-I'_\lambda(x) \,\text{as}\,n\to +\infty.
\]
 We show that $I'_\lambda(x)=0$. For every 
$\zeta\in C_0^\infty (\mathbb{R},\mathbb{R}^N)$, we have
 $$ 
I'_\lambda(x) \zeta=\lim_{n\to \infty} I'_\lambda(x_n) \zeta=0.
$$
 Consequently, $I'_\lambda(x)=0$ and \eqref{eq3.6} holds.
\end{proof}

\begin{lemma}\label{lem3.2}
Suppose that $f\in L^{2}$ and {\rm  (A3), (A4), (A6), (A8)}
 are satisfied. Then, there exists $\lambda_0>0$ such that any bounded $(PS)$
sequence of $I_{\lambda}$ has a convergent subsequence when $\lambda > \lambda_0$.
\end{lemma}

\begin{proof}
 Let $(x_{n})$ be a bounded sequence such that $(I_{\lambda}(x_{n}) )$ is
 bounded and $I'_{\lambda}(x_{n})\to 0$ as $n\to \infty$. Then, after 
passing to a subsequence, we have $x_{n}\rightharpoonup x$ in 
$X^{\alpha}_{\lambda}$ and $y_{n}\to 0$ in $L^{2}({\{L(t)<b\}})$ where 
$y_{n}:=x_{n}-x$. Moreover,
 \begin{equation}\label{eq3.13}
 \|y_{n}\|_{L^{2}}^{2}
\leq {1\over {\lambda b}} \int_{\{L\geq b\}}\lambda L(t)y_{n} \cdot  y_{n}dt
 + \int_{\{L< b\}}|y_{n}|^{2}dt
\leq {1\over{\lambda b }}\|y_{n}\|^{2}_{\lambda} +o(1).
 \end{equation}
Setting $N_{2}:=\sup_{n}\|y_{n}\|_{L^{\infty}}$. By (A4), we obtain
\[
|\widetilde{W}(t,y_{n})|
=\big|\frac12 \nabla W(t,y_{n})\cdot y_{n}-W(t,y_{n})\big|
\leq \max_{|u|\in [0,N_{2}]}\overline{W}(u)(N_{2}+1), \quad \forall n,
\]
which, jointly with \eqref{eq3.13} and (A8) yields 
\begin{equation}\label{eq3.14}
\begin{aligned}
\int_{|y_{n}|\geq R_1}\nabla W(t,y_{n})\cdot y_{n}dt 
&\leq c_1 \int_{|y_{n}|\geq R_1}\widetilde{W}(t,y_{n})|y_{n}|^{2-\beta}dt\\
&\leq  c c_1 R^{-\beta}_1\int_{|y_{n}|\geq R_1}|y_{n}|^{2}dt \\
&\leq  {c c_1\over{\lambda b }}\|y_{n} \|^{2}_{\lambda} +o(1).
\end{aligned}
\end{equation}
Furthermore, using (A4), \eqref{eq3.9} and \eqref{eq3.13}, we have
\begin{equation}\label{eq3.15}
\begin{aligned}
&\int_{|y_{n}|< R_1}\nabla W(t,y_{n})y_{n}dt \\
&\leq \int_{|y_{n}|\leq \sigma }\varepsilon |y_{n}|^{2}dt
 + \int_{\sigma <|y_{n}|< R_1}|\nabla W(t,y_{n})||y_{n}|dt\\
&\leq \varepsilon \int_{|y_{n}|\leq \delta } |y_{n}|^{2}dt 
 + \max_{|u|\in [\sigma,R_1]}\overline{W}(u)\sigma^{-1}\int_{\mathbb{R}} |y_{n}|^{2}dt \\ 
&\leq  {c \over{\lambda b }}\|y_{n} \|^{2}_{\lambda} +o(1).
\end{aligned}
\end{equation}
Because $f\in L^{2}$, one has, for any $\varepsilon>0$, there exists 
$T_{\varepsilon}>0$ such that
\[ 
\Big(\int_{|t|\geq T_{\varepsilon}}|f(t)|^{2}dt\Big)^{1/2} < \varepsilon.
\]
 Using \eqref{eq2.8} and the H\"older inequality, we have
\begin{equation}\label{eq3.16}
\big|\int_{|t|\geq T_{\varepsilon}}f(t)y_{n}dt\big|
\leq \Big(\int_{|t|\geq T_{\varepsilon}}|f(t)|^{2}dt\Big)^{1/2}
 \Big(\int_{\mathbb{R}}|y_{n}|^{2}dt\Big)^{1/2} 
\leq c \varepsilon\quad  \forall n.
 \end{equation}
Obviously
\begin{equation}\label{eq3.17}
 \int_{|t|< T_{\varepsilon}}f(t)\cdot y_{n}dt
\leq \Big(\int_{\mathbb{R}}|f(t)|^{2}dt\Big)^{1/2}
 \Big(\int_{|t|<T_{\varepsilon}}|y_{n}|^{2}dt\Big)^{1/2}\to 0,
\end{equation}
 as $ n\to \infty$. By \eqref{eq3.16} and \eqref{eq3.17}, we have
\begin{equation}\label{eq3.18}
\int_{\mathbb{R}}f(t)\cdot y_{n}(t)dt\to 0,
\end{equation}
as $n\to \infty$. Consequently, a combination of \eqref{eq3.4}, \eqref{eq3.14}, 
\eqref{eq3.15} and \eqref{eq3.18} implies 
\begin{align*}
o(1)= I'_{\lambda}(y_{n})y_{n}
&=\|y_{n}\|^{2}_{\lambda}-\int_{\mathbb{R}}\nabla W(t,y_{n})\cdot y_{n}dt
 +\int_{\mathbb{R}}f(t)\cdot y_{n}dt\\
&\geq  (1-{c c_1\over{\lambda b}}-{c \over{\lambda b}})\|y_{n}\|^{2}_{\lambda} 
+o(1).
\end{align*}
Choosing $\lambda_0>0$ large enough such the term
$(1-{c c_1\over{\lambda b}}-{c \over{\lambda b}})$ is positive.
 When $\lambda> \lambda_0$, we obtain $y_{n}\to 0$ and then
 $x_{n}\to x$ in $X^{\alpha}_{\lambda}$.
\end{proof}

\begin{lemma}\label{lem3.3}
If $f\in L^{2}$ and {\rm (A3), (A4), (A6)--(A8)} are satisfied, 
then $I_{\lambda}$ satisfies the $(PS)$ condition whenever $\lambda>\lambda_0$.
\end{lemma}

\begin{proof}
 Let $(x_{n})$ be a $(PS)$ sequence of $I_{\lambda}$. 
By Lemma \ref{lem3.2}, it suffices to prove that $(x_{n})$ is bounded. 
Indeed, assume that $\|x_{n}\|_{\lambda} \to \infty$ as $n\to \infty$ 
and setting $y_{n}:=\frac {x_{n}}{\|x_{n}\|_{\lambda}}$. 
Then $\|y_{n}\|_{\lambda} =1$ and $\|y_{n}\|_{L^{p}}\leq \delta_{p}$ 
for $p\in [2,+\infty]$. Moreover, we have
\[
o(1)={I'_{\lambda}(x_{n})x_{n}\over{\|x_{n} \|^{2}_{\lambda} }}
= 1-\int_{\mathbb{R}}{\nabla W(t,x_{n})\cdot x_{n}\over{\|x_{n}\|^{2}_{\lambda }}}dt+o(1),
\]
as $n\to \infty$. We obtain
\begin{equation}\label{eq3.19}
\int_{\mathbb{R}}{\nabla W(t,x_{n})\cdot y_{n}\over{|x_{n}|}} |y_{n}|dt 
= \int_{\mathbb{R}}{\nabla W(t,x_{n})\cdot x_{n}\over{\|x_{n} \|^{2}_{\lambda}}}dt \to 1,
\end{equation}
as $ n\to \infty$. Let $0\leq \alpha_1<\alpha_{2}$ and 
$ \omega^{\alpha_1,\alpha_{2}}_{n}:=\{t\in \mathbb{R}; \alpha_1\leq |x_{n}(t)|<\alpha_{2} \}$.
 By \eqref{eq2.8} and because $(x_{n})$ is a $(PS)$ sequence of $I_{\lambda}$, 
then there exists $N_0>0$ such that for $n\geq N_0$ we have
\begin{align*}
c+\|x_{n}\|_{\lambda} 
&\geq I_{\lambda}(x_{n})-\frac12 I'_{\lambda}(x_{n})x_{n}  \\
&\geq \int_{\mathbb{R}}\widetilde{W}(t,x_{n})dt+\frac12 \int_{\mathbb{R}}f(t)\cdot x_{n}dt\\
& \geq \int_{\mathbb{R}}\widetilde{W}(t,x_{n})dt-{\delta_{2}\over{2}}
\|f\|_{L^{2}}\|x_{n}\|_{\lambda} .
\end{align*}
This implies, for $n\geq N_0$, that
\begin{equation}\label{eq3.20}
\begin{aligned}
&c(1+\|x_{n}\|_{\lambda} )  \\
&\geq \int_{\mathbb{R}}\widetilde{W}(t,x_{n})dt \\
&= \int_{\omega^{0,\alpha_1}_{n}}\widetilde{W}(t,x_{n})dt
+\int_{\omega^{\alpha_1,\alpha_{2}}_{n}}\widetilde{W}(t,x_{n})dt
+ \int_{\omega^{\alpha_{2},\infty}_{n}}\widetilde{W}(t,x_{n})dt.
\end{aligned}
\end{equation}
By (A3), for any $\varepsilon >0 (\varepsilon < {1\over{3}})$ there 
exists $\kappa_{\varepsilon}>0$ such that
$$ 
|\nabla W(t,u)|\leq ({\varepsilon\over {\delta^{2}_{2}}})|u|, \quad
\forall |u|\leq \kappa_{\varepsilon} , t\in \mathbb{R}.
 $$
Thus,
\begin{equation}\label{eq3.21}
\int_{\omega^{0,\kappa_{\varepsilon}}}{|\nabla W(t,x_{n})|\over{|x_{n}|}} 
|y_{n}|^{2}dt
 \leq \int_{\omega^{0,\kappa_{\varepsilon}}} 
{\varepsilon\over { \delta^{2}_{2}}}|y_{n}|^{2}dt 
\leq {\varepsilon\over { \delta^{2}_{2}}}\|y_{n}\|_{L^{2}}^{2}
\leq \varepsilon, \quad \forall n.
\end{equation}
Because $\beta >1$ and by (A8), \eqref{eq2.8} and \eqref{eq3.20} we can choose 
$\theta_{\varepsilon}\geq R_1$ large enough such that
 \begin{equation}\label{eq3.22}
 \begin{aligned}
 \int_{ \omega^{\theta_{\varepsilon},+\infty} }
 \frac{\nabla W(t,x_{n})x_{n}}{\|x_{n} \|^{2}_{\lambda}}dt
& \leq  \int_{ \omega^{\theta_{\varepsilon},+\infty} }c_1 
 \frac{| y_{n}| \widetilde{W}(t,x_{n})}{ |x_{n}|^{\beta-1}\|x_{n}\|_{\lambda}} dt
 \\
 & \leq  c_1 \|y_{n}\|_{L^{\infty}} \int_{ \omega^{\theta_{\varepsilon},+\infty} }
 \frac{\widetilde{W}(t,x_{n})}{\theta^{\beta-1}_{\varepsilon}\|x_{n}\|_{\lambda}}
 \\
 &\leq \frac{ c c_1 \|y_{n}\|_{L^{\infty}}(1+\|x_{n}\|_{\lambda}) }
 {\theta^{\beta-1}_{\varepsilon}\|x_{n}\|_{\lambda}}
 \\
 &\leq  {c \delta_{\infty}\over{ \theta^{\beta-1}_{\varepsilon}}}< \varepsilon,\quad 
\forall n\geq N_0.
 \end{aligned}
 \end{equation}
 By (A7), we have
 $ \widetilde{W}(t,x_{n}(t)) \geq C^{\theta_{\varepsilon}}_{\kappa_{\varepsilon}}
|x_{n}|^{2}$ for $ t\in \omega^{\kappa_{\varepsilon},\theta_{\varepsilon}}_{n}$. 
Noting
 $C^{\theta_{\varepsilon}}_{\kappa_{\varepsilon}}>0$ it follows from \eqref{eq3.20} 
that
 \begin{equation}
 \begin{aligned}
 \int_{\omega^{\kappa_{\varepsilon},\theta_{\varepsilon}}}|y_{n}|^{2}dt 
&= {1\over { \|x_{n} \|^{2}_{\lambda} }}\int_{\omega^{\kappa_{\varepsilon},
 \theta_{\varepsilon}}}|x_{n}|^{2}dt\\
&\leq {1\over {C^{\theta_{\varepsilon}}_{\kappa_{\varepsilon}}\|x_{n} 
 \|^{2}_{\lambda} }} \int_{\omega^{\kappa_{\varepsilon},
 \theta_{\varepsilon}}}\widetilde{W}(t,x_{n})dt \\
&\leq  {c(1+\|x_{n}\|_{\lambda} )
 \over{ C^{\theta_{\varepsilon}}_{\kappa_{\varepsilon}}\|x_{n} \|^{2}_{\lambda} }}
\to 0,
 \end{aligned}
\end{equation}
as $ n\to +\infty$, which yields that
 \begin{equation}\label{eq3.24}
\int_{\omega^{\kappa_{\varepsilon},\theta_{\varepsilon}}}
 {|\nabla W(t,x_{n})|\over{|x_{n}|}}|y_{n}|^{2}dt 
\leq \tau_{\varepsilon} \int_{\omega^{\kappa_{\varepsilon},
 \theta_{\varepsilon}}} |y_{n}|^{2}dt\to 0,
\end{equation}
as $n\to \infty$, where 
$\tau_{\varepsilon}=\max_{|u|\in [\kappa_{\varepsilon},\theta_{\varepsilon}]}
\overline{W}(u)\cdot\kappa_{\varepsilon}$. Hence, by \eqref{eq3.21}, 
\eqref{eq3.22} and \eqref{eq3.24}, we have
 \begin{equation*}
 \int_{\mathbb{R}}{\nabla W(t,x_{n})\cdot y_{n}\over{|x_{n}|}}|y_{n}|
\leq \int_{\mathbb{R}}{|\nabla W(t,x_{n})|\over{|x_{n}|}}|y_{n}|^{2} 
\leq 3\varepsilon < 1,
 \end{equation*}
 for $n$ large enough, a contradiction with \eqref{eq3.19} and then 
$(x_{n})$ is bounded in $X^{\alpha}_{\lambda}$. 
 \end{proof}

 \begin{lemma}\label{lem3.4}
 If {\rm (A3)} holds and $f\in L^{2}$, then there exist $\rho, \gamma, f_0 >0$
such that $ I_{\lambda}(x)_{/\|x\|_{\lambda} =\rho}\geq \gamma$ when 
$\|f\|_{L^{2}}< f_0$.
 \end{lemma}

\begin{proof} 
By (A3), for $\varepsilon:={1\over{4\delta^{2}_{2}}}$ there exists 
$\sigma_1=\sigma_1(\varepsilon)$ such that
 \begin{equation}\label{eq3.25}
 |W(t,x)|\leq \varepsilon |x|^{2}, \forall t \in \mathbb{R}, |x|\leq \sigma_1.
 \end{equation}
Thus, for $\|x\|_{\lambda} \leq \rho := \sigma_1/\delta_{\infty}$,
  by \eqref{eq3.25}, we obtain
 \begin{equation*}
 I_{\lambda}(x)\geq \frac12 \|x \|^{2}_{\lambda} -\varepsilon \int_{\mathbb{R}}|x|^{2}dt
-\|f\|_{L^{2}}\|x\|_{L^{2}}\geq \|x\|_{\lambda} \big({1\over{4}}\|x\|_{\lambda} 
-\|f\|_{L^{2}}\delta_{2}\Big).
 \end{equation*}
 Let $\gamma:=\rho ({1\over{4\delta_{2}}}\rho-\|f\|_{L^{2}}\delta_{2})$. 
Then, if $\|f\|_{L^{2}}< f_0:={1\over{4\delta^{2}_{2}}}\rho$, we have
$I_{\lambda}(x)_{/\|x\|_{\lambda} =\rho}\geq \gamma$.
 \end{proof}

 \begin{lemma} 
If $\|f\|_{L^{2}}< f_0$ and $ (A3), (A7)$ are satisfied, then there
exists $x_1\in X^{\alpha}_{\lambda}\backslash\{0\}$ such that
$I'_{\lambda}(x_1)=0$.
\end{lemma}

\begin{proof} 
Since $f\in L^{2}\backslash\{0\}$, we can choose 
$\xi \in X^{\alpha}_{\lambda}$ such that $\int_{\mathbb{R}}f(t)\cdot\xi(t)dt<0$. 
By (A3) and (A7) we have $W\geq0$ and
\begin{equation*}
I_{\lambda}(s\xi)\leq {s^{2}\over{2}}\|\xi\|^{2}_{\lambda}
+s \int_{\mathbb{R}}f(t)\cdot\xi(t)dt < 0,
 \end{equation*}
 for $s$ small enough.
Thus $C_1:=\inf\{I_{\lambda}(x),x\in \overline{B}_{\rho}(0)\}<0$, where $\rho$ 
is the constants given by Lemma \ref{lem3.4}. From Ekeland's variational principle
there exists a sequence $(x_{n})\subset \overline{B}_{\rho}$ such that
 $C_1\leq I_{\lambda}(x_{n})<C_1+{1\over{n}}$. Then, by a standard procedure,
 we can show that $(x_{n})\subset X^{\alpha}_{\lambda}$ is bounded 
$(PS)$ sequence. Consequently, Lemma \ref{lem3.2} implies that, there exist 
$x_1\in X^{\alpha}_{\lambda}$ such that 
$x_{n}\to x_1\in X^{\alpha}_{\lambda},I'_{\lambda}(x_1)=0$ and 
$I_{\lambda}(x_1)=C_1<0$ when $ \lambda > \lambda_0$.
\end{proof}

\subsection{Proof of Theorem \ref{thm1.2}}
Let $h(s)=s^{-2}W(t,sx_0)$ for $t\in \mathbb{R}, s>0$. Then, by (A7),
 \begin{equation*}
 h'(s)=s^{-3}[-2W(t,s x_0)+\nabla W(t,sx_0).sx_0]>0,\quad\text{for }
 t\in \mathbb{R}, \; s>0.
 \end{equation*}
Integrating the above from $1$ to $\eta$, we obtain
\begin{equation}\label{eq3.26}
W(t,\eta x_0)\geq \eta^{2} W(t,x_0),\quad\text{for } t\in \mathbb{R},\; \eta>1.
\end{equation}
From \eqref{eq3.26}, we have for $s>1$,
 \begin{align*}
I_{\lambda}(s x_0)
&=\int_{\mathbb{R}}(\lambda s^{2} L(t)x_0\cdot x_0- W(t,sx_0 ))dt
 +s \int_{\mathbb{R}}f(t)\cdot x_0 dt \\
&\leq s^{2}(\int_{\mathbb{R}}\lambda L(t)x_0\cdot x_0- W(t,x_0 )dt)
 +s \int_{\mathbb{R}}f(t)\cdot x_0 dt.
\end{align*}
Let
$$
e(t)=\begin{cases}
sx_0, &\text{if }  t \in [-T_0,T_0]\\
0,& \text{if } t \in \mathbb{R} \backslash [-T_0,T_0],
\end{cases}
$$
By (A9) there exists $s_0\geq 1$ such that $\|e \|_{\lambda}> \rho$ and 
$I_{\lambda}(e)<0$.
Since $I_{\lambda}(0)=0$ and all the assumptions of Lemma \ref{lem2.8} are 
satisfied, so $I_{\lambda}$ possesses a critical point 
$x_{2}\in X^{\alpha}_{\lambda}$ with $I'_{\lambda}(x_{2})=0$ and 
$I_{\lambda}(x_{2})=C_{2}>0$ whenever $\lambda> \lambda_0$. 

\subsection{Proof of Corollary \ref{coro1.3}}
If (A9') holds, let $e\in C_0^{\infty}(\mathbb{R})\backslash\{0\}$. Then, 
by Fatou's Lemma and by $W\geq 0$ we have
 $$
I_{\lambda}(se)\leq s^2[ \frac{1}{2}\|e\|_{\lambda}^2
-\int_{e \neq 0}\frac{W(t,se)}{(se)^2}e^2\, dt]
+s \int_{\mathbb{R}}f(t).e(t) dt\to -\infty
$$
 as $s\to +\infty$, which implies that $I_{\lambda}(se)<0$ for $s>0$ large.
 Combining this with Lemmas \ref{lem3.3} and \ref{lem3.4}, all the assumptions 
of Lemma \ref{lem2.8} are satisfied, so $I_{\lambda}$ possesses a critical point 
$x_3\in X^{\alpha}_{\lambda}$ with $I'_{\lambda}(x_3)=0$ and $I_{\lambda}(x_3)>0$
 whenever $\lambda> \lambda_0$. 

\section{An example}

 Let  $L(t)=h(t)I_{N}$ where
\begin{gather*}
h(t)=\begin{cases}
0, &\text{if } |t|<1,\\
2n^{2}|t-n|, &\text{if }  |t|\geq1, |t-n|\leq {1\over{2n^{2}}}\;
 (n\in \mathbb{Z}, |n|\geq 1),\\
1,&\text{elsewhere,}
\end{cases} \\
 W(t,x)= k(t) |x|^{2} \ln (1+|x|^{2}), 
\end{gather*}
where $k:\mathbb{R}\to \mathbb{R}^{+}$ is a continuous bounded function 
with $\inf k(t)>0$.
A straightforward computation shows that $L$ and $W$ satisfies 
Theorem \ref{thm1.2} and Corollary \ref{coro1.3} but they do not satisfy 
the corresponding results on the above papers, in particular $W$ do not satisfy the
 Ambrosetti- Rabinowitz Condition (A5).


\subsection*{Acknowledgments}
The author would like to thank the referee for their careful reading,
 critical comments and helpful suggestions, which helped to improve the
 quality of this article. 


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\end{document}