Ground state solutions for a class of fractional Hamiltonian systems

Abstract

In this paper we study the following 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. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and $W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})$ is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.