Ricerche di Matematica

, Volume 68, Issue 2, pp 727–743

# Ground state solutions for a class of fractional Hamiltonian systems

Article

## 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.

## Keywords

Fractional Hamiltonian systems Fractional Sobolev space Ground state solution Critical point theory Concentration phenomena

## Mathematics Subject Classification

34C37 35A15 35B38

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