Existence of infinitely many solutions for a class of fractional Hamiltonian systems

Abstract

In this paper we study solutions of the following nonperiodic fractional Hamiltonian systems: $$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$ where $$\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}$$ and $$_{t}D^{\alpha }_{\infty }$$ are the left and right Liouville-Weyl fractional derivatives of order $$\alpha $$ on the whole axis $$\mathbb {R}$$ respectively, $$L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}$$ and $$W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}$$ are suitable functions. Applying a new Fountain Theorem established by W. Zou, we prove the existence of infinitely many solutions for the above system in the case where the matrix L(t) is not required to be either uniformly positive definite or coercive, and $$W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})$$ is superquadratic at infinity in x but does not needed to satisfy the Ambrosetti-Rabinowitz condition.