Homoclinic Solutions for Fractional Hamiltonian Systems with Indefinite Conditions

Abstract

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\R,\R^n),
\end{array}
\right.
\eqno(\mbox{FHS})
$$
where $\alpha\in (1/2,1)$, $t\in \R$, $u\in \R^n$, $L\in C(\R,\R^{n^2})$ is a symmetric matrix for all $t\in \R$, $W\in C^1(\R\times\R^n,\R)$ and $\nabla W(t,u)$
is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, when $L(t)$ is allowed to be indefinite and $W(t,u)$ satisfies some new superquadratic conditions, we show that (FHS) possesses infinitely many homoclinic solutions via a variant fountain
theorem. Recent results are generalized and significantly improved.