On fractional Hamiltonian systems with indefinite sign sub-quadratic potentials

Abstract

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems \begin{eqnarray*} \begin{cases} _tD_\infty^\alpha(_{-\infty}D_t^\alpha u(t))+L(t)u(t) =\nabla F(t, u(t)),\\ u\in H^\alpha(\mathbb R,\mathbb R^n), \end{cases} \end{eqnarray*} where $\alpha \in (1/2, 1), u\in {\mathbb{R}}^{N}$, $L\in C({\mathbb{R}}, {\mathbb{R}}^{N\times N})$ and $F\in C^1({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$. The novelty of this paper is that, under the relaxed assumptions on $F(t, x)$ and $L(t)$, we obtain infinitely many solutions via genus properties in critical point theory. Recent results are generalized and significantly improved.