Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} {t}D{\infty}^{\alpha}({-\infty}D{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t))\ u\in H^{\alpha}($\mathbb{R}$, $\mathbb{R}$^{N}).\nonumber \end{eqnarray} where $\alpha \in (1\mathrm{/}2,1)$, $t\in \mathbb{R}$, $u\in $\mathbb{R}$^{n}$,$L\in C($\mathbb{R}$, $\mathbb{R}$^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}($\mathbb{R}$\times $\mathbb{R}$^{n}, $\mathbb{R}$)$,and$\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $l\in C(\mathbb{R},\mathbb{R})$ such that $(L(t)u,u)\ge l(t)\mathrm{\mid }u{\mathrm{\mid }}^2$ for all $t\in \mathbb{R}$, $u\in {\mathbb{R}}^n$ and the following conditions on $l$: ${\inf }_{t\in \mathbb{R}}l(t)>0$ and there exists $r_0>0$ such that, for any $M>0$ $$m(\{t\in (y-r_0,y+r_0)\mathrm{/}l(t)\le M\})\to 0as\mathrm{\mid }y\mathrm{\mid }\to \mathrm{\infty }\mathrm{.}$$ are satisfied and $W$ is of subquadratic growth as $\mathrm{\mid }u\mathrm{\mid }\to +\mathrm{\infty }$, we show that (\ref{eq00}) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Math. Methods Appl. Sci., DOI: 10.1002/mma.3031] are significantly improved.