Homoclinic solutions for fractional Hamiltonian systems via variational method

Abstract

We study the multiplicity of weak nonzero solutions for fractional Hamiltonian systems of the form $$_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha} u(t)) +L(t)u(t)=a(t)\nabla V(t,u(t)),\quad t\in \mathbb{R},$$ where $\alpha\in (1/2,1]$, $_{-\infty}D_{t}^{\alpha}$ and $_{t}D_{\infty}^{\alpha}$ are left and the right Liouville-Weyl fractional derivatives of order $\alpha$ on real line $\mathbb{R}$, $L(t)$ is a positive defined symmetric $n\times n$ matrix and $V:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}$ satisfies specific growth conditions. A result is proved using variational method and the generalized Clark's theorem. Some recent results are extended and improved.