Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

Abstract

In this article, we consider the following fractional Hamiltonian systems:

$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$

where $\alpha \in (1/2, 1)$, $\lambda >0$ is a parameter, $L\in C(\mathbb {R}, \mathbb {R}^{n\times n})$ and $W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})$. Unlike most other papers on this problem, we require 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 $\mathbb {I}$ of $\mathbb {R}$. Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on $\mathbb {R} \setminus \mathbb {I}$ as $\lambda \rightarrow \infty ,$ and converge to $\tilde{u}$ in $H^{\alpha }(\mathbb {R})$; here $\tilde{u} \in E_{0}^{\alpha }$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb {I}$. Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.