Existence of solution for Liouville-Weyl Fractional Hamiltonian systems

Abstract

In this paper, we investigate the existence of solution for the following fractional Hamiltonian systems:
\begin{eqnarray}\label{eq00}
_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t))\\
u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}).\nonumber
\end{eqnarray}
where $\alpha \in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$ and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $l\in C(\mathbb{R}, \mathbb{R})$ such that $(L(t)u,u)\geq l(t)|u|^{2}$ for all $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$ and the following conditions on $l$: $\inf_{t\in \mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that, for any $M>0$
$$
m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) \to 0\;\;\mbox{as}\;\;|y|\to \infty.
$$
are satisfied and $W$ is superquadratic growth as $|u| \to +\infty$, we show that (\ref{eq00}) possesses at least one nontrivial solution via mountain pass theorem. Recent results in \cite{CT} are significantly improved. We do not assume that $l(t)$ have a limit for $|t| \to \infty$.