Existence of solution for Liouville-Weyl Fractional Hamiltonian systems

摘要

In this paper, we investigate the existence of solution for the following fractional Hamiltonian systems:egin{eqnarray}label{eq00}_{t}D_{infty}^{lpha}(_{-infty}D_{t}^{lpha}u(t)) + L(t)u(t) = & abla W(t,u(t))uin H^{lpha}(mathbb{R}, mathbb{R}^{N}).onumberend{eqnarray}where $lpha in (1/2, 1)$, $tin mathbb{R}$, $uin mathbb{R}^{n}$, $Lin C(mathbb{R}, mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $tin mathbb{R}$, $Win C^{1}(mathbb{R}imes mathbb{R}^{n}, mathbb{R})$ and $abla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $lin C(mathbb{R}, mathbb{R})$ such that $(L(t)u,u)geq l(t)|u|^{2}$ for all $tin mathbb{R}$, $uin mathbb{R}^{n}$ and the following conditions on $l$: $inf_{tin mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that, for any $M>0$ $$ m({tin (y-r_{0}, y+r_{0})/;;l(t)leq M}) o 0;;mbox{as};;|y|o infty. $$ are satisfied and $W$ is superquadratic growth as $|u| o +infty$, we show that (ef{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 $|t| o infty$.