Existence and concentration of solution for a fractional Hamiltonian systems with positive semi-definite matrix

Abstract

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^α_{\infty}(_{-\infty}D^α_{t}u(t))-λL(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^α(\mathbb{R},\mathbb{R}^n), \end{array} \right. \eqno(\mbox{FHS})_λ$$ where $\alpha \in (1/2,1)$, $t\in \mathbb{R}$, $u\in {\mathbb{R}}^n$, $\lambda >0$ is a parameter, $L\in C(\mathbb{R},{\mathbb{R}}^{n^2})$ is a symmetric matrix for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R}\times {\mathbb{R}}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, 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 $\tilde{u}\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^n)$. Here, $\tilde{u}\in E_0^{\alpha }$ is a solution of 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 $\alpha =1$.