Existence of solutions to fractional Hamiltonian systems with local superquadratic conditions

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. For more information see https://ejde.math.txstate.edu/Volumes/2020/29/abstr.html