Concentration of ground state solution for a fractional Hamiltonian Systems

Abstract

In this paper we are concerned with the existence of ground states solutions for the following fractional Hamiltonian systems $$\{\begin{array}{l}{-}_t{D}_{\infty }^{\alpha }{(}_{-\infty }{D}_t^{\alpha }u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\ u\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^n),\end{array}(FHS{)}_{\lambda }$$ 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 $t\in \mathbb{R}$, $W\in C^1(\mathbb{R}\times {\mathbb{R}}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. 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 Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (FHS)${}_{\lambda }$ has a ground sate solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to $u\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^n)$, where $u\in E_0^{\alpha }$ is a ground state solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Recent results are generalized and significantly improved.