Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials

Abstract

In this paper, we prove the existence of infinitely many solutions for the following nonperiodic fractional Hamiltonian system

$$\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in {\mathbb {R}}\\ u\in H^{\alpha }({\mathbb {R}}), \end{array}\right. \end{aligned}$$

where $_{-\infty }D_{t}^{\alpha }$ and $_{t}D^{\alpha }_{\infty }$ are left and right Liouville–Weyl fractional derivatives of order $\frac{1}{2}<\alpha <1$ on the whole axis, respectively, $L\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})$ is a symmetric matrix valued function unnecessary coercive and $W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})$ is only locally defined and superquadratic near the origin with respect to x. Our results extend and improve some existing results in the literature.