Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 29, pp. 1-12.

Title: Existence of solutions to fractional Hamiltonian systems with 
       local superquadratic conditions

Authors: Zijun Guo    (Jiangxi Normal Univ., Nanchang, China)
         Qingye Zhang (Jiangxi Normal Univ., Nanchang, China)

Abstract:
 In this article, we study the existence of solutions for the fractional
 Hamiltonian system
 $$\displaylines{
  {}_tD_\infty^\alpha(_{-\infty}D_t^\alpha u(t))+L(t)u(t)=\nabla W(t,u(t)),\cr
  u\in H^\alpha(\mathbb{R},\mathbb{R}^N),
 }$$
 where $ {}_tD_\infty^\alpha$ and $_{-\infty}D_t^\alpha$ are the Liouville-Weyl
 fractional derivatives of order $1/2<\alpha<1$,
 $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ is a symmetric matrix-valued function,
 which is unnecessarily required to be coercive,
 and $W\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$ satisfies some kind of
 local superquadratic conditions, which is rather weaker than the usual
 Ambrosetti-Rabinowitz condition.
 
Submitted September 21, 2019. Published April 06, 2020.
Math Subject Classifications: 26A33, 35A15, 35B38, 37J45.
Key Words: Fractional Hamiltonian system; variational method; superquadratic.