Fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems$$left{ begin{array}{ll} - _tD^{alpha}_{infty}(_{-infty}D^{alpha}_{t}u(t))-lambda L(t)u(t)+nabla W(t,u(t))=0,[0.1cm] uin H^{alpha}(mathbb{R},mathbb{R}^n), end{array}right. ~~~~~~~~~~~~~~~~~(FHS)_lambda$$where $alphain (1/2,1)$, $tin mathbb{R}$, $uin mathbb{R}^n$, $lambda>0$ is a parameter, $Lin C(mathbb{R},mathbb{R}^{n^2})$ is a symmetric matrix, $Win C^1(mathbb{R} times mathbb{R}^n,mathbb{R})$. Assuming that$L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)equiv 0$ is allowed to occur in some finite interval $T$ of $mathbb{R}$,$W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_lambda$ has a solution which vanishes on$mathbb{R}setminus T$ as $lambda to infty$, and converges to some $tilde{u}in H^{alpha}(R, R^n)$. Here, $tilde{u}in E_{0}^{alpha}$ is a solutionof the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $alpha =1$.