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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 259, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/259\hfil Existence of solutions]
{Existence of solutions for fractional Hamiltonian systems}

\author[C. Torres \hfil EJDE-2013/259\hfilneg]
{C\'esar Torres}  % in alphabetical order

\address{C\'esar Torres \newline
Departamento de Ingenier\'{\i}a  Matem\'atica and
Centro de Modelamiento Matem\'atico,
UMR2071 CNRS-UChile,
Universidad de Chile,
Santiago, Chile}
\email{ctorres@dim.uchile.cl}

\thanks{Submitted June 10, 2013. Published November 26, 2013.}
\subjclass[2000]{26A33, 34C37, 35A15, 35B38}
\keywords{Liouville-Weyl fractional derivative; fractional Hamiltonian systems;
critical point; variational methods}

\begin{abstract}
 In this work we  prove the existence of solutions for the
 fractional differential equation
 $$
 _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t)
 =  \nabla W(t,u(t)),\quad 
 u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}).
 $$
 where $\alpha \in (1/2, 1)$.
 Assuming $L$ is coercive at infinity we show that this equation
 has at least one nontrivial solution.
\end{abstract}

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


\section{Introduction}

Fractional differential equations both ordinary and partial ones are
applied in mathematical modeling of processes in physics, mechanics,
control theory, biochemistry, bioengineering and economics.
Therefore the theory of fractional differential equations is an area
intensively developed during last decades
\cite{OAJTMJS,RH,RMJK,JSOAJTM,BWMBPG}.
The monographs \cite{AKHSJT,KMBR,IP}, enclose a review of methods
of solving fractional differential equations, which are an extension
of processes from differential equations theory.

Recently, also equations including both - left and right fractional derivatives,
are discussed. Let us point out that according to integration by parts
formulas in fractional calculus, we obtain equations mixing
left and right operators. Apart from their possible applications,
equations with left and right derivatives are an interesting and
new field in fractional differential equations theory.
Some works in this topic can be founded in papers
\cite{TABS,DBJT,MK} and their references.

Recently Jiao and Zhou \cite{FJYZ}, for the first time,
showed that the critical point theory is an effective approach for
studying the existence for the following fractional boundary-value problem
\begin{equation}\label{Eq01}
\begin{gathered}
_{t}D_{T}^{\alpha}({_{0}D_{t}^{\alpha}}u(t))
=  \nabla F(t,u(t)),\quad\text{a.e. }t\in [0,T],\\
u(0)  =  u(T) = 0,
\end{gathered}
\end{equation}
and obtained the existence of at least one nontrivial solution.

Motivated by this work, in this paper we  consider a fractional
differential equation with left and right fractional derivatives
 on $\mathbb{R}$, that is,
\begin{equation} \label{Eq02}
_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t)
=  \nabla W(t,u(t))
\end{equation}
where $\alpha \in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$,
$L\in C(\mathbb{R}, \mathbb{R}^{n\times n})$ is a symmetric
matrix-valued function and $W:\mathbb{R}\times \mathbb{R}^{n} \to \mathbb{R}$;
satisfies the following conditions:
\begin{itemize}
\item[(L1)] $L(t)$ is positive definite symmetric matrix for all
 $t\in \mathbb{R}$ and there exists an $l\in C(\mathbb{R}, (0,\infty))$
such that $l(t) \to +\infty$ as $t \to \infty$ and
    \begin{equation}\label{Eq03}
    (L(t)x,x) \geq l(t)|x|^2,\quad\text{for all }
t\in \mathbb{R}\; x\in \mathbb{R}^{n}.
    \end{equation}

\item[(W1)] $W\in C^{1}(\mathbb{R} \times \mathbb{R}^{n}, \mathbb{R})$
and there exists a constant $\mu >2$ such that
$$
0< \mu W(t,x) \leq (x, \nabla W(t,x)),\quad\text{for all }
t\in \mathbb{R}\; x\in\mathbb{R}^{n}\setminus \{0\}.
$$

\item[(W2)] $|\nabla W(t,x)| = o(|x|)$ as $x\to 0$ uniformly with respect
to $t\in \mathbb{R}$.

\item[(W3)] There exists $\overline{W} \in C(\mathbb{R}^{n}, \mathbb{R})$
such that
$$
|W(t,x)| + |\nabla W(t,x)| \leq |\overline{W(x)}|
\quad\text{for every }x\in \mathbb{R}^{n}\; t\in \mathbb{R}.
$$
\end{itemize}


In particular, if $\alpha = 1$, Equation \eqref{Eq02} reduces to the
standard second-order differential equation
\begin{equation}\label{HEq01}
u'' - L(t)u + \nabla W(t,u)=0,
\end{equation}
where $W: \mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}$
is a given function and $\nabla W(t,u)$ is the gradient of $W$ at $u$.
The existence of homoclinic solution is one of the most important
problems in the history of that kind of equations, and has been
studied intensively by many mathematicians.
Assuming that $L(t)$ and $W(t,u)$ are independent of $t$, or
$T$-periodic in $t$, many authors have studied the existence
of homoclinic solutions for \eqref{HEq01} via critical point
theory and variational methods. In this case, the existence of
 homoclinic solution can be obtained by going to the limit of periodic
solutions of approximating problems.

If $L(t)$ and $W(t,u)$ are neither autonomous nor periodic in $t$,
this problem is quite different from the ones just described, because
the lack of compacteness of the Sobolev embedding. In \cite{PRKT} the
authors considered \eqref{HEq01} without periodicity assumptions
on $L$ and $W$ and showed that \eqref{HEq01} possesses one homoclinic
solution by using a variant of the mountain pass theorem without the
 Palais-Smale contidion. In \cite{WOMW}, under the same assumptions
of \cite{PRKT}, the authors, by employing a new compact embedding theorem,
obtained the existence of homoclinic solution of \eqref{HEq01}.

Physical models containing left and right fractional differential operators
have recently renewed attention from scientists which is mainly due to
applications as models for physical phenomena exhibiting anomalous diffusion.
 A strong motivation for investigating the fractional differential equation
\eqref{Eq02} comes from symmetry fractional advection-dispersion
equation (SADE for short). A fractional advection-dispersion equation
(ADE for short) is a generalization of the classical ADE in which the
second-order derivative is replaced with a fractional-order derivative.
In contrast to the classical ADE, the fractional ADE has solutions that
resemble the highly skewed and heavy-tailed breakthrough curves observed
in field and laboratory studies \cite{DBSWMM1}, \cite{DBRSMM}, in particular
in contaminant transport of ground-water flow \cite{DBSWMM2}.
In \cite{DBSWMM2}, the authors state that solutes moving through a highly
heterogeneous aquifer violations violates the basic assumptions of local
second order theories because of large deviations from the stochastic
process of Brownian motion.

According to \cite{DBSWMM1}, the one-dimensional form of the fractional
 ADE can be written as
\begin{equation}\label{Eq02-1}
\frac{\partial \mathcal{C}}{\partial t}
= -v\frac{\partial \mathcal{C}}{\partial x}
+ \mathcal{D}j \frac{\partial^{\gamma} \mathcal{C}}{\partial^{\gamma} x}
+ \mathcal{D}(1-j) \frac{\partial^{\gamma} \mathcal{C}}{\partial (-x)^{\gamma}},
\end{equation}
where $\mathcal{C}$ is the expected concentration, $t$ is time, $v$ is a
constant mean velocity, $x$ is the distance in the direction of mean velocity,
$\mathcal{D}$ is a constant dispersion coefficient,
$0 \leq j \leq 1$ describes the skewness of the transport process, and
$\gamma$ is the order of left and right fractional differential operators.
For discussions of this equation, see \cite{DBSWMM2}

A special case of the fractional ADE \eqref{Eq02-1} describes symmetric
transitions, where $j = 1/2$. Defining the symmetric operator equivalent
to the Riesz potential \cite{SSAKOM}
\begin{equation}\label{Eq02-2}
2\nabla^{\gamma} = D_{+}^{\gamma} + D_{-}^{\gamma}
\end{equation}
gives the mass balance equation for advection and symmetric fractional dispersion
\begin{equation}\label{Eq02-3}
\frac{\partial \mathcal{C}}{\partial t} = - v \nabla \mathcal{C}
+ \mathcal{D}\nabla^{\gamma} \mathcal{C}.
\end{equation}
The fractional ADE has been studied in one dimension (\cite{DBSWMM2}),
over infinite domains by using the Fourier transform of fractional
differential operators to determine a classical solution. Variational methods,
especially the Galerkin approximation has been investigated to find the
solutions of fractional BVP \cite{JFJR} and fractional ADE \cite{VEJR}
on a finite domain by establishing some suitable fractional derivative spaces.

Our goal in this paper is to show how variational methods based on
Mountain pass theorem can be used to get existence results for \eqref{Eq02}.
However, the direct application of the mountain pass theorem is not enough
since the Palais-Smale sequences might lose compactness in the whole
space $\mathbb{R}$. To overcome this difficulty we proof a version of
compact embedding for fractional space following the ideas of \cite{WOMW}.
Before stating our results let us introduce the main ingredients involved
in our approach. We define
$$
\|u\|_{I_{-\infty}^{\alpha}}^2
= \int_{-\infty}^{\infty} |u(t)|^2dt
+ \int_{-\infty}^{\infty}|_{-\infty}D_{t}^{\alpha}u(t)|^2dt
$$
and the space
$$
I_{-\infty}^{\alpha}(\mathbb{R})
= \overline{C_{0}^{\infty}(\mathbb{R}, \mathbb{R}^{n})}^{\|\cdot\|_{\alpha}}.
$$
Now we say that $u\in I_{-\infty}^{\alpha}(\mathbb{R})$ is a weak solution
of \eqref{Eq02} if
$$
\int_{-\infty}^{\infty} [(_{-\infty}D_{t}^{\alpha}u(t),
{_{-\infty}}D_{t}^{\alpha}v(t)) + (L(t)u(t),v(t))]dt
= \int_{-\infty}^{\infty} (\nabla W(t,u(t)), v(t))dt,
$$
for all $v\in I_{-\infty}^{\alpha}(\mathbb{R})$.
For $u\in I_{-\infty}^{\alpha}(\mathbb{R})$ we may define the functional
\begin{equation}\label{Eq03b}
I(u) = \frac{1}{2}\int_{-\infty}^{\infty}[|_{-\infty}D_{t}^{\alpha}u(t)|^2
+ (L(t)u(t),u(t))]dt - \int_{-\infty}^{\infty}W(t,u(t))dt.
\end{equation}
which is of class $C^1$. We say that $u\in E^{\alpha}$ is a weak solution
of \eqref{Eq02} if $u$ is a critical point of $I$.

Now we are in a position to state our main existence theorem.

\begin{theorem}\label{tm01}
Suppose that {(L1), (W1)--(W3)} hold. Then \eqref{Eq02} possesses
at least one nontrivial solution.
\end{theorem}


The rest of the paper is organized as follows:
in section 2, subsection 2.1, we describe the Liouville-Weyl fractional
calculus; in subsection 2.2 we introduce the fractional space that we
use in our work and some proposition are proven which will aid in our analysis.
In section 3, we will prove Theorem \ref{tm01}.

\section{Preliminary results}

\subsection{Liouville-Weyl Fractional Calculus}

The Liouville-Weyl fractional integrals of order $0<\alpha < 1$ are defined as
\begin{gather}\label{LWeq01}
_{-\infty}I_{x}^{\alpha}u(x)
= \frac{1}{\Gamma (\alpha)} \int_{-\infty}^{x}(x-\xi)^{\alpha - 1}u(\xi)d\xi,\\
\label{LWeq02}
_{x}I_{\infty}^{\alpha}u(x) = \frac{1}{\Gamma (\alpha)}
 \int_{x}^{\infty}(\xi - x)^{\alpha - 1}u(\xi)d\xi\,.
\end{gather}
The Liouville-Weyl fractional derivative of order $0<\alpha <1$ are defined
as the left-inverse operators of the corresponding Liouville-Weyl fractional
integrals
\begin{gather}\label{LWeq03}
_{-\infty}D_{x}^{\alpha}u(x) = \frac{d }{d x} {_{-\infty}}I_{x}^{1-\alpha}u(x),\\
\label{LWeq04}
_{x}D_{\infty}^{\alpha}u(x) = -\frac{d }{d x} {_{x}}I_{\infty}^{1-\alpha}u(x)\,.
\end{gather}
The definitions \eqref{LWeq03} and \eqref{LWeq04} may be written in an
alternative form:
\begin{gather}\label{LWeq05}
_{-\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,\\
\label{LWeq05b}
_{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}

We establish the Fourier transform properties of the fractional integral
and fractional differential operators. Recall that the Fourier
transform $\widehat{u}(w)$ of $u(x)$ is defined by
$$
\widehat{u}(w) = \int_{-\infty}^{\infty} e^{-ix.w}u(x)dx.
$$
Let $u(x)$ be defined on $(-\infty, \infty)$. Then the Fourier
transform of the Liouville-Weyl integral and differential operator satisfies
\begin{gather}\label{LWeq06}
\widehat{ _{-\infty}I_{x}^{\alpha}u(x)}(w) = (iw)^{-\alpha}\widehat{u}(w),\\
\label{LWeq07}
\widehat{ _{x}I_{\infty}^{\alpha}u(x)}(w) = (-iw)^{-\alpha}\widehat{u}(w),\\
\label{LWeq08}
\widehat{ _{-\infty}D_{x}^{\alpha}u(x)}(w) = (iw)^{\alpha}\widehat{u}(w),\\
\label{LWeq09}
\widehat{ _{x}D_{\infty}^{\alpha}u(x)}(w) = (-iw)^{\alpha}\widehat{u}(w)
\end{gather}

\subsection{Fractional derivative spaces}

In this section we introduce some fractional spaces for more
detail see \cite{VEJR}.
Let $\alpha > 0$. Define the semi-norm
$$
|u|_{I_{-\infty}^{\alpha}} = \|_{-\infty}D_{x}^{\alpha}u\|_{L^2}
$$
and the norm
\begin{equation}\label{FDEeq01}
\|u\|_{I_{-\infty}^{\alpha}}
= \Big( \|u\|_{L^2}^2 + |u|_{I_{-\infty}^{\alpha}}^2 \Big)^{1/2}\,.
\end{equation}
Let
$$
I_{-\infty}^{\alpha} (\mathbb{R})
= \overline{C_{0}^{\infty}(\mathbb{R})}^{\|\cdot\|_{I_{-\infty}^{\alpha}}}.
$$
Now we define the fractional Sobolev space $H^{\alpha}(\mathbb{R})$
in terms of the fourier transform. Let $0< \alpha < 1$, let the semi-norm
\begin{equation}\label{FDEeq02}
|u|_{\alpha} = \||w|^{\alpha}\widehat{u}\|_{L^2}
\end{equation}
and norm
$$
\|u\|_{\alpha} = \left( \|u\|_{L^2}^2 + |u|_{\alpha}^2 \right)^{1/2},
$$
and let
$$
H^{\alpha}(\mathbb{R}) = \overline{C_{0}^{\infty}(\mathbb{R})}^{\|\cdot\|_{\alpha}}.
$$
We note a function $u\in L^2(\mathbb{R})$ belongs to
$I_{-\infty}^{\alpha}(\mathbb{R})$ if and only if
\begin{equation}\label{FDEeq03}
|w|^{\alpha}\widehat{u} \in L^2(\mathbb{R}).
\end{equation}
Especially,
\begin{equation}\label{FDEeq04}
|u|_{I_{-\infty}^{\alpha}} = \||w|^{\alpha}\widehat{u}\|_{L^2}.
\end{equation}
Therefore $I_{-\infty}^{\alpha}(\mathbb{R})$ and
$H^{\alpha}(\mathbb{R})$ are equivalent with equivalent
semi-norm and norm. Analogous to $I_{-\infty}^{\alpha}(\mathbb{R})$
we introduce $I_{\infty}^{\alpha}(\mathbb{R})$. Let the semi-norm
$$
|u|_{I_{\infty}^{\alpha}} = \|_{x}D_{\infty}^{\alpha}u\|_{L^2}
$$
and the norm
\begin{equation}\label{FDEeq05}
\|u\|_{I_{\infty}^{\alpha}}
= \Big( \|u\|_{L^2}^2 + |u|_{I_{\infty}^{\alpha}}^2 \Big)^{1/2}.
\end{equation}
Let
$$
I_{\infty}^{\alpha}(\mathbb{R})
= \overline{C_{0}^{\infty}(\mathbb{R})}^{\|\cdot\|_{I_{\infty}^{\alpha}}}.
$$
Moreover $I_{-\infty}^{\alpha}(\mathbb{R})$ and
$I_{\infty}^{\alpha}(\mathbb{R})$ are equivalent, with equivalent semi-norm
and norm \cite{VEJR}.

Now we give the prove of the Sobolev lemma.

\begin{theorem}\label{FDEtm01}
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{FDEeq06}
\sup_{x\in \mathbb{R}} |u(x)| \leq C \|u\|_{\alpha}
\end{equation}
\end{theorem}

\begin{proof} By the Fourier inversion theorem, if
$\widehat{u} \in L^{1}(\mathbb{R})$, then $u$ is continuous and
$$
\sup_{x\in \mathbb{R}} |u(x)| \leq \|\widehat{u}\|_{L^{1}}.
$$
Hence, to prove the theorem it is sufficient to prove that
$$
\|\widehat{u}\|_{L^{1}} \leq \|u\|_{\alpha},
$$
so by Schwarz inequality, we have
\begin{align*}
\int_{\mathbb{R}} |\widehat{u}(w)|dw
& =  \int_{\mathbb{R}} (1 + |w|^2)^{\alpha /2}|\widehat{u}(w)|
 \frac{1}{(1+|w|^2)^{\alpha /2}}dw\\
& \leq  \Big( \int_{\mathbb{R}} (1 + |w|^{2\alpha}) |\widehat{u}(w)|^2dw
\Big)^{1/2}\Big( \int_{\mathbb{R}}(1+|w|^2)^{-\alpha}dw \Big)^{1/2}.
\end{align*}
The first integral on the right is $\|u\|_{\alpha}^2$, so the theorem
depends on the fact that
$$
\int_{\mathbb{R}} (1 + |w|^2)^{-\alpha}dw < \infty
$$
precisely when $\alpha > 1/2$.
\end{proof}

\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
$$
\int_{\mathbb{R}} |u(x)|^{q}dx \leq \|u\|_{\infty}^{q-2}\|u\|_{L^2}^2\,.
$$
\end{remark}

Now we introduce a new fractional space. Let
$$
X^{\alpha} = \big\{ u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{n}):
\int_{\mathbb{R}} |_{-\infty}D_{t}^{\alpha}u(t)|^2 + L(t)u(t).u(t) dt < \infty
 \big\}\,.
$$
The space $X^{\alpha}$ is a Hilbert space with the inner product
$$
\langle u,v \rangle_{X^{\alpha}}
= \int_{\mathbb{R}} (_{-\infty}D_{t}^{\alpha}u(t) , \; _{-\infty}D_{t}^{\alpha}v(t)) + L(t)u(t).v(t)dt
$$
and the corresponding norm
$$
\|u\|_{X^{\alpha}}^2 = \langle u,u \rangle_{X^{\alpha}}
$$

\begin{lemma}\label{FDElm01}
Suppose $L$ satisfies {\rm (L1)}. Then $X^{\alpha}$ is continuously embedded
in $H^{\alpha}(\mathbb{R},\mathbb{R}^{n})$.
\end{lemma}

\begin{proof}
 Since $l\in C(\mathbb{R}, (0,\infty))$ and $l$ is coercive,
then $l_{\rm min} = \min_{t\in \mathbb{R}}l(t)$ exists, so we have
$$
(L(t)u(t) , u(t)) \geq l(t)|u(t)|^2 \geq l_{\rm min}|u(t)|^2,\quad
\forall t\in \mathbb{R}.
$$
Then
\begin{align*}
l_{\rm min}\|u\|_{\alpha}^2
&=  l_{\rm min}\Big( \int_{\mathbb{R}} |_{-\infty}D_{t}^{\alpha}u(t)|^2
 + |u(t)|^2dt\Big)\\
 &\leq  l_{\rm min} \int_{\mathbb{R}}|_{-\infty}D_{t}^{\alpha}u(t)|^2dt
 + \int_{\mathbb{R}}(L(t)u(t),u(t))dt
\end{align*}
So
\begin{equation}\label{FDEeq07}
\|u\|_{\alpha}^2 \leq K \|u\|_{X^{\alpha}}^2
\end{equation}
where $K = \max \{l_{\rm min}, 1\}/l_{\rm min}$.
\end{proof}

\begin{lemma}\label{FDElm02}
Suppose $L$ satisfies {\rm (L1)}. Then the imbedding of $X^{\alpha}$
in $L^2(\mathbb{R})$ is compact.
\end{lemma}

\begin{proof}
 We note first that by Lemma \ref{FDElm01} and Remark \ref{FDEnta01} we have
$$
X^{\alpha} \hookrightarrow L^2(\mathbb{R})\text{ is continuous}.
$$
Now, let $(u_k) \in X^{\alpha}$ be a sequence such that
$u_k \rightharpoonup u$ in $X^{\alpha}$. We will show that $u_k \to u$
in $L^2(\mathbb{R})$. Suppose, without loss of generality, that
$u_k \to 0$ in $X^{\alpha}$. The Banach-Steinhaus theorem implies that
$$
A = \sup_k\|u_k\|_{X^{\alpha}} < +\infty
$$
Let $\epsilon >0$; there is $T_{0}<0$ such that $\frac{1}{l(t)} \leq \epsilon$
for all $t$ such that $t\leq T_{0}$. Similarly, there is $T_1>0$,
such that $\frac{1}{l(t)}\leq \epsilon$ for all $t\geq T_1$.
 Sobolev's theorem (see e.g. \cite{CAS}) implies that $u_k \to 0$
uniformly on $\overline{\Omega} = [T_{0}, T_1]$, so there is a
$k_{0}$ such that
\begin{equation}\label{FDEeq08}
\int_{\Omega} |u_k(t)|^2dt \leq \epsilon,\quad\text{for all }k\geq k_{0}.
\end{equation}
Since $1/l(t)\leq \epsilon$ on $(-\infty , T_{0}]$ we have
\begin{equation}\label{FDEeq09}
\int_{-\infty}^{T_{0}} |u_k(t)|^2dt
\leq \epsilon \int_{-\infty}^{T_{0}} l(t)|u_k(t)|^2dt \leq \epsilon A^2.
\end{equation}
Similarly, since $1/l(t) \leq \epsilon$ on $[T_1, +\infty)$, we have
\begin{equation}\label{FDEeq10}
\int_{T_1}^{+\infty} |u_k(t)|^2dt \leq \epsilon A^2.
\end{equation}
Combining \eqref{FDEeq08}, \eqref{FDEeq09} and \eqref{FDEeq10} we obtain
 $u_k \to 0$ in $L^2(\mathbb{R}, \mathbb{R}^{n})$.
\end{proof}

\begin{lemma}\label{FDE-lem01}
There are constants $c_1>0$ and $c_2>0$ such that
\begin{gather}\label{FDE-eq01}
W(t,u) \geq c_1|u|^{\mu},\quad |u|\geq 1\,,\\
\label{FDE-eq02}
W(t,u) \leq c_2|u|^{\mu},\quad |u|\leq 1\,.
\end{gather}
\end{lemma}

\begin{proof} By (W1) we note that
$$
\mu W(t,\sigma u) \leq (\sigma u, \nabla W(t, \sigma u)).
$$
Let $f(\sigma) = W(t,\sigma u)$, then
\begin{equation}\label{FDE-eq03}
\frac{d}{d\sigma}\big( f(\sigma) \sigma^{-\mu} \big)\geq 0\,.
\end{equation}
Now we consider two cases

\noindent\textbf{Case 1.}
$|u|\leq 1$. In this case we integrate \eqref{FDE-eq03}, from $1$
to $1/|u|$ and we obtain
\begin{equation}\label{FDE-eq04}
W(t,u) \leq W(t,\frac{u}{|u|})|u|^{\mu}.
\end{equation}

\noindent\textbf{Case 2.} $|u|\geq 1$. In this case we integrate
\eqref{FDE-eq03}, from $1/|u|$ to $1$ and we obtain
\begin{equation}\label{FDE-eq05}
W(t,u) \geq |u|^{\mu}W(t, \frac{u}{|u|}).
\end{equation}
Now, since $u\in \mathbb{R}^{n}$, $\frac{u}{|u|}\in B(0,1)$.
So, since $W$ is continuous and $B(0,1)$ is compact, there are $c_1>0$
and $c_2>0$ such that
$$
c_1 \leq W(t,u) \leq c_2,\quad\text{for every }u\in B(0,1).
$$
Therefore, we get the statement of the lemma.
\end{proof}

\begin{remark}\label{FDE-nta01} \rm
By lemma \ref{FDE-lem01}, we have
\begin{equation}\label{FDE-eq06}
W(t, u) = o(|u|^2)\quad \text{as $u\to 0$ uniformly in }t\in\mathbb{R}
\end{equation}
In addition, by {\rm (W2)},  for any $u\in \mathbb{R}^{n}$ such that
$|u| \leq M_1$, there exists some constant $d>0$ (dependent on $M_1$)
such that
\begin{equation}\label{FDE-eq07}
|\nabla W(t,u(t))| \leq d|u(t)|\,.
\end{equation}
\end{remark}

As in \cite[lemma 2]{WOMW}, we obtain the following result.

\begin{lemma}\label{FDElm03}
Suppose that {\rm (L1), (W1)-(W2)} are satisfied.
If $u_k \rightharpoonup u$ in $X^{\alpha}$, then
$\nabla W(t, u_k) \to \nabla W(t, u)$ in
$L^2(\mathbb{R}, \mathbb{R}^{n})$.
\end{lemma}

\begin{proof} Assume that $u_k \rightharpoonup u$ in $X^{\alpha}$.
Then there exists a constant $d_1>0$ such that, by Banach-Steinhaus
theorem and \eqref{FDEeq06},
$$
\sup_{k\in \mathbb{N}} \|u_k\|_{\infty} \leq d_1,\quad
\|u\|_{\infty} \leq d_1.
$$
By (W2), for any $\epsilon >0$ there is $\delta >0$ such that
$$
|u_k|< \delta\quad\text{implies}\quad |\nabla W(t,u_k)|\leq \epsilon |u_k|\,.
$$
By (W3) there is $M>0$ such that
$$
|\nabla W(t,u_k)| \leq M,\;\;\text{forall}\;\delta < u_k \leq d_1\,.
$$
Therefore, there exists a constant $d_2>0$ (dependening on $d_1$) such that
$$
|\nabla W(t, u_k(t))| \leq d_2|u_k(t)|,\;\;|\nabla W(t,u(t))| \leq d_2|u(t)|
$$
for all $k\in \mathbb{N}$ and $t\in \mathbb{R}$. Hence,
$$
|\nabla W(t,u_k(t)) - \nabla W(t,u(t))| \leq d_2(|u_k(t)| + |u(t)|)
\leq d_2(|u_k(t) - u(t)| + 2|u(t)|)\,.
$$
Since, by lemma \ref{FDElm02}, $u_k \to u$ in $L^2(\mathbb{R}, \mathbb{R}^{n})$,
passing to a subsequence if necessary, it can be assumed that
$$
\sum_{k=1}^{\infty} \|u_k - u\|_{L^2} < \infty\,.
$$
But this implies $u_k(t) \to u(t)$ almost everywhere
$t\in \mathbb{R}$ and
$$
\sum_{k=1}^{\infty}|u_k(t) - u(t)| = v(t) \in L^2(\mathbb{R},\mathbb{R}^{n})\,.
$$
Therefore,
$$
|\nabla W(t,u_k(t)) - \nabla W(t,u(t))| \leq d_2(v(t) + 2|u(t)|)\,.
$$
Then, using the Lebesgue's convergence theorem, the lemma is proved.
\end{proof}

Now we introduce more symbols and some definitions.
Let $\mathfrak{B}$ be a real Banach space,
$I\in C^{1}(\mathfrak{B},\mathbb{R})$, which means that $I$ is a
continuously Fr\'echet-differentiable functional defined on $\mathfrak{B}$.
Recall that $I\in C^{1}(\mathfrak{B},\mathbb{R})$ is said to satisfy
the (PS) condition if any sequence $\{u_k\}_{k\in \mathbb{N}} \in \mathfrak{B}$,
for which $\{I(u_k)\}_{k\in \mathbb{N}}$ is bounded and $I'(u_k) \to 0$
as $k\to +\infty$, possesses a convergent subsequence in $\mathfrak{B}$.

Moreover, let $B_{r}$ be the open ball in $\mathfrak{B}$ with the radius
$r$ and centered at $0$ and $\partial B_{r}$ denote its boundary.
We obtain the existence of solutions to \eqref{Eq02} by use of
the following well-known Mountain Pass Theorems, see \cite{PR}.

\begin{theorem}\label{FDEtm02}
Let $\mathfrak{B}$ be a real Banach space and 
$I\in C^{1}(\mathfrak{B}, \mathbb{R})$ satisfying the (PS) condition. 
Suppose that $I(0) = 0$ and
\begin{itemize}
\item[(i)] There are constants $\rho , \beta >0$ such that 
$I|_{\partial B_{\rho}} \geq \beta$, and
\item[(ii)] There is and $e\in \mathfrak{B} \setminus \overline{B_{\rho}}$ 
such that $I(e)\leq 0$.
\end{itemize}
Then $I$ possesses a critical value $c\geq \beta$. Moreover $c$ can be 
characterized as
$$
c = \inf_{\gamma \in \Gamma} \max_{s\in [0,1]} I(\gamma (s)),
$$
where
$$
\Gamma = \{\gamma \in C([0,1] , \mathfrak{B}):\gamma (0) = 0,\;\gamma (1) = e\}
$$
\end{theorem}

\section{Proof of Theorem \ref{tm01}}
Now we establish the corresponding variational framework to obtain 
the existence of solutions for \eqref{Eq02}. 
Define the functional $I: X^{\alpha} \to \mathbb{R}$ by
\begin{equation} \label{TMeq01}
\begin{aligned}
I(u) &=  \int_{\mathbb{R}} \Big[ \frac{1}{2}|_{-\infty}D_{t}^{\alpha}u(t)|^2
  + \frac{1}{2}(L(t)u(t),u(t)) - W(t,u(t))\Big]dt \\
     &=  \frac{1}{2}\|u\|_{X^{\alpha}}^2 - \int_{\mathbb{R}} W(t,u(t))dt
\end{aligned}
\end{equation}

\begin{lemma}\label{TMlm01}
Under the conditions of Theorem \ref{tm01}, we have
\begin{equation}\label{TMeq02}
I'(u)v = \int_{\mathbb{R}} 
[ (_{-\infty}D_{t}^{\alpha}u(t), _{-\infty}D_{t}^{\alpha}v(t)) 
+ (L(t)u(t),v(t)) - (\nabla W(t,u(t)),v(t)) ]dt
\end{equation}
for all $u,v \in X^{\alpha}$, which yields 
\begin{equation}\label{TMeq03}
I'(u)u = \|u\|_{X^{\alpha}}^2 - \int_{\mathbb{R}}(\nabla W(t,u(t)), u(t))dt.
\end{equation}
Moreover, $I$ is a continuously Fr\'echet-differentiable functional 
defined on $X^{\alpha}$; i.e., $I\in C^{1}(X^{\alpha}, \mathbb{R})$.
\end{lemma}

\begin{proof}
 We firstly show that $I: X^{\alpha} \to \mathbb{R}$. 
By \eqref{FDE-eq06}, there is a $\delta >0$ such that $|u| \leq \delta$ 
implies 
\begin{equation}\label{TMeq04}
W(t,u) \leq \epsilon |u|^2\quad\text{for all }t\in \mathbb{R}
\end{equation}
Let $u\in X^{\alpha}$, then $u \in C(\mathbb{R},\mathbb{R}^{n})$, 
the space of continuous function $u\in \mathbb{R}$ such that 
$u(t)\to 0$ as $|t| \to +\infty$. Therefore there is a constant
 $R>0$ such that $|t| \geq R$ implies $|u(t)|\leq \delta$. 
Hence, by \eqref{TMeq04}, we have
\begin{equation}\label{TMeq05}
\int_{\mathbb{R}} W(t, u(t)) \leq \int_{-R}^{R} W(t,u(t))dt 
+ \epsilon \int_{|t|\geq R}|u(t)|^2dt < +\infty.
\end{equation}
Combining \eqref{TMeq01} and \eqref{TMeq05}, we show that
 $I:X^{\alpha} \to \mathbb{R}$.

Now we prove that $I\in C^{1}(X^{\alpha}, \mathbb{R})$. Rewrite $I$ as follows
$$
I = I_1 - I_2,
$$
where
$$
I_1 = \frac{1}{2} \int_{\mathbb{R}} [|_{-\infty}D_{t}^{\alpha}u(t)|^2
 + (L(t)u(t),u(t))]dt,\quad
I_2 = \int_{\mathbb{R}}W(t,u(t))dt
$$
It is easy to check that $I_1 \in C^{1}(X^{\alpha},\mathbb{R})$ and
\begin{equation}\label{TMeq06}
I'_1(u)v = \int_{\mathbb{R}}\left[ (_{-\infty}D_{t}^{\alpha}u(t), _{-\infty}D_{t}^{\alpha}v(t)) + (L(t)u(t), v(t))\right]dt.
\end{equation}
Thus it is sufficient to show this is the case for $I_2$. 
In the process we will see that
\begin{equation}\label{TMeq07}
I'_2(u)v = \int_{\mathbb{R}}(\nabla W(t,u(t)), v(t))dt,
\end{equation}
which is defined for all $u,v\in X^{\alpha}$. For any given 
$u\in X^{\alpha}$, let us define $J(u): X^{\alpha} \to \mathbb{R}$ as follows
$$
J(u)v = \int_{\mathbb{R}} (\nabla W(t,u(t)), v(t))dt,\quad 
\forall v\in X^{\alpha}.
$$
It is obvious that $J(u)$ is linear. Now we show that $J(u)$ is bounded. 
Indeed, for any given $u\in X^{\alpha}$, by \eqref{FDE-eq07}, 
there is a constant $d_3>0$ such that
$$
|\nabla W(t,u(t))| \leq d_3|u(t)|,
$$
which yields that, by  H\"older's inequality and lemma \ref{FDElm01},
\begin{equation}\label{TMeq08}
\begin{aligned}
|J(u)v| &=  \big| \int_{\mathbb{R}} (\nabla W(t, u(t)), v(t)) dt\big| \\
&\leq d_3\int_{\mathbb{R}}|u(t)||v(t)|dt 
\leq  \frac{d_3}{l_{\rm min}}\|u\|_{X^{\alpha}}\|v\|_{X^{\alpha}}.
\end{aligned}
\end{equation}
Moreover, for $u$ and $v \in X^{\alpha}$, by mean value theorem, we have
$$
\int_{\mathbb{R}} W(t,u(t) + v(t))dt - \int_{\mathbb{R}} W(t, u(t))dt
 = \int_{\mathbb{R}} (\nabla W(t,u(t) + h(t)v(t)))dt,
$$
where $h(t)\in (0,1)$. Therefore, by lemma \ref{FDElm02} and  H\"older's
 inequality, we have
\begin{equation} \label{TMeq09}
\begin{aligned}
&\int_{\mathbb{R}} (\nabla W(t,u(t) + h(t)v(t)), v(t))dt
- \int_{\mathbb{R}} (\nabla W(t, u(t)), v(t))dt \\
&= \int_{\mathbb{R}} (\nabla W(t,u(t)) + h(t)v(t)
- \nabla W(t,u(t)), v(t))dt \to 0
\end{aligned}
\end{equation}
as $v\to 0$ in $X^{\alpha}$. Combining \eqref{TMeq08} and \eqref{TMeq09},
we see that \eqref{TMeq07} holds. It remains to prove that $I'_2$ is continuous.
Suppose that $u \to u_{0}$ in $X^{\alpha}$ and note that
\begin{align*}
\sup_{\|v\|_{X^{\alpha}} = 1} |I'_2(u)v - I'_2(u_{0})v|
&=  \sup_{\|v\|_{X^{\alpha}}= 1}
 \big| \int_{\mathbb{R}} (\nabla W(t,u(t))
 - \nabla W(t,u_{0}(t)), v(t))dt \big|\\
&\leq  \sup_{\|v\|_{X^{\alpha}} = 1} \|\nabla W(.,u(.))
 - \nabla W(.,u_{0}(.))\|_{L^2}\|v\|_{L^2}\\
&\leq  \frac{1}{\sqrt{l_{\min}}} \|\nabla W(.,u(.))
 - \nabla W(.,u_{0}(.))\|_{L^2}
\end{align*}
By lemma \ref{FDElm02}, we obtain that $I'_2(u)v - I'_2(u_{0})v \to 0$
as $\|u\|_{X^{\alpha}} \to \|u_{0}\|_{X^{\alpha}}$ uniformly with
respect to $v$, which implies the continuity of $I'_2$ and
$I\in C^{1}(X^{\alpha}, \mathbb{R})$.
\end{proof}

\begin{lemma}\label{TMlm02}
Under  conditions {\rm (L1), (W1), (W2)}, $I$ satisfies the (PS) condition.
\end{lemma}

\begin{proof} 
Assume that $(u_k)_{k\in \mathbb{N}} \in X^{\alpha}$ is a sequence such 
that $\{I(u_k)\}_{k\in \mathbb{N}}$ is bounded and $I'(u_k) \to 0$ 
as $k \to +\infty$. Then there exists a constant $C_1>0$ such that
\begin{equation}\label{TMeq10}
|I(u_k)|\leq C_1,\quad \|I'(u_k)\|_{(X^{\alpha})^{*}} \leq C_1
\end{equation}
for every $k\in \mathbb{N}$.
We firstly prove that $\{u_k\}_{k\in \mathbb{N}}$ is bounded in 
$X^{\alpha}$. By \eqref{TMeq01}, \eqref{TMeq03} and ($W_1$), we have
\begin{equation} \label{TMeq11}
\begin{aligned}
C_1 + \|u_k\|_{X^{\alpha}}
& \geq  I(u_k)-\frac{1}{\mu}I'(u_k)u_k \\
&= \left( \frac{\mu}{2}  - 1\right)\|u_k\|_{X^{\alpha}}^2 
- \int_{\mathbb{R}} [W(t,u_k(t)) - \frac{1}{\mu}(\nabla W(t,u_k(t)), u_k(t))]dt \\
&\geq  \left( \frac{\mu}{2}  - 1\right)\|u_k\|_{X^{\alpha}}^2.
\end{aligned}
\end{equation}
Since $\mu > 2$, the inequality \eqref{TMeq11} shows that
$\{u_k\}_{k\in \mathbb{N}}$ is bounded in $X^{\alpha}$.
So passing to a subsequence if necessary, it can be assumed that
 $u_k \rightharpoonup u$ in $X^{\alpha}$ and hence, by lemma \ref{FDElm02},
$u_k \to u$ in $L^2(\mathbb{R},\mathbb{R}^{n})$.
It follows from the definition of $I$ that
\begin{equation} \label{TMeq12}
\begin{aligned}
&(I'(u_k) - I'(u))(u_k - u)   \\
&= \|u_k - u\|_{X^{\alpha}}^2 - \int_{\mathbb{R}}[\nabla W(t,u_k)
- \nabla W(t,u)](u_k - u)dt.
\end{aligned}
\end{equation}
Since $u_k \to u$ in $L^2(\mathbb{R}, \mathbb{R}^{n})$, we have
(see lemma \ref{FDElm03}) $\nabla W(t, u_k(t)) \to \nabla W(t,u(t))$
in $L^2(\mathbb{R}, \mathbb{R}^{n})$. Hence
$$
\int_{\mathbb{R}} ( \nabla W(t,u_k(t))-\nabla W(t,u(t)), u_k(t)-u(t) )dt \to 0
$$
as $k\to +\infty$. So \eqref{TMeq12} implies
$\|u_k - u\|_{X^{\alpha}} \to 0$ as $k\to +\infty$.
\end{proof}

Now we are in the position to give the proof of Theorem \ref{tm01}.
 We divide the proof into several steps.


\begin{proof}[Proof of theorem \ref{tm01}]
\textbf{Step 1.} It is clear that $I(0) = 0$ and 
$I\in C^{1}(X^{\alpha}, \mathbb{R})$ satisfies the (PS) 
condition by lemma \ref{TMlm01} and \ref{TMlm02}.

\noindent
\textbf{Step 2.} Now We show that there exist constant $\rho >0$ 
and $\beta >0$ such that $I$ satisfies the condition (i) of 
theorem \ref{FDEtm02}. By lemma \ref{FDElm02}, there is a $C_{0}> 0$ 
such that
$$
\|u\|_{L^2} \leq C_{0} \|u\|_{X^{\alpha}}.
$$
On the other hand by theorem \ref{FDEtm01}, there is $C_{\alpha}>0$ such that
$$
\|u\|_{\infty} \leq C_{\alpha} \|u\|_{X^{\alpha}}.
$$
By \eqref{FDE-eq06}, for all $\epsilon >0$, there exists $\delta >0$ such that
$$
W(t,u(t)) \leq \epsilon |u(t)|^2\quad \text{wherever }|u(t)| < \delta.
$$
Let $\rho = \frac{\delta}{C_{\alpha}}$ and $\|u\|_{X^{\alpha}} \leq \rho$; 
we have $\|u\|_{\infty} \leq \frac{\delta}{C_{\alpha}}.C_{\alpha} = \delta$. 
Hence
$$
|W(t,u(t))| \leq \epsilon |u(t)|^2\quad\text{for all }t\in \mathbb{R}.
$$
Integrating on $\mathbb{R}$, we obtain
$$
\int_{\mathbb{R}}W(t,u(t))dt \leq \epsilon \|u\|_{L^2}^2 
\leq \epsilon C_{0}^2\|u\|_{X^{\alpha}}^2\,.
$$
So, if $\|u\|_{X^{\alpha}} = \rho$, then
$$
I(u) = \frac{1}{2}\|u\|_{X^{\alpha}}^2 - \int_{\mathbb{R}}W(t,u(t))dt 
\geq (\frac{1}{2} - \epsilon C_{0}^2)\|u\|_{X^{\alpha}}^2 
= (\frac{1}{2} - \epsilon C_{0}^2)\rho^2.
$$
And it suffices to choose $\epsilon = \frac{1}{4C_{0}^2}$ to obtain
\begin{equation}\label{TMeq13}
I(u) \geq \frac{\rho^2}{4C_{0}^2} = \beta >0\,.
\end{equation}

\noindent
\textbf{Step 3.} It remains to prove that there exists an
 $e\in X^{\alpha}$ such that $\|e\|_{X^{\alpha}}> \rho$ and 
$I(e)\leq 0$, where $\rho$ is defined in Step 2. Consider
$$
I(\sigma u) = \frac{\sigma^2}{2}\|u\|_{X^{\alpha}}^2 
- \int_{\mathbb{R}}W(t,\sigma u(t))dt
$$
for all $\sigma \in \mathbb{R}$. By \eqref{FDE-eq01}, there is $c_1>0$ 
such that
\begin{equation}\label{TMeq14}
W(t,u(t)) \geq c_1|u(t)|^{\mu}\quad\text{for all } |u(t)| \geq 1.
\end{equation}
Take some $u\in X^{\alpha}$ such that $\|u\|_{X^{\alpha}} = 1$. 
Then there exists a subset $\Omega$ of positive measure of $\mathbb{R}$ 
such that $u(t) \neq 0$ for $t\in \Omega$. Take $\sigma >0$ such that 
$\sigma |u(t)| \geq 1$ for $t\in \Omega$. Then by \eqref{TMeq14}, we obtain
\begin{equation}\label{TMeq15}
I(\sigma u) \leq \frac{\sigma^2}{2} - c_1\sigma^{\mu} 
\int_{\Omega}|u(t)|^{\mu} dt.
\end{equation}
Since $c_1>0$ and $\mu >2$, \eqref{TMeq15} implies that $I(\sigma u) <0$ 
for some $\sigma >0$ with $\sigma |u(t)|\geq 1$ for $t\in \Omega$ and 
$\|\sigma u\|_{X^{\alpha}}> \rho$, where $\rho$ is defined in Step 2. 
By theorem \ref{FDEtm02}, $I$ possesses a critical value $c\geq \beta >0$ 
given by
$$
c = \inf_{\gamma \in \Gamma}\max_{s\in [0,1]}I(\gamma (s)),
$$
where
$$
\Gamma = \{\gamma \in C([0,1], X^{\alpha}): \gamma (0) = 0,\;\gamma (1) = e\}.
$$
Hence there is $u\in X^{\alpha}$ such that
$I(u) = c$, $I'(u) = 0$.
\end{proof}

\subsection*{Acknowledgements}
The author  was  partially supported by grant 0607 from MECESUP.

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