Existence of solutions to fractional Hamiltonian systems with combined nonlinearities

Abstract

This article concerns the existence of solutions for the fractional Hamiltonian system $$\displaylines{ - _tD^{\alpha}_{\infty}\big(_{-\infty}D^{\alpha}_{t}u(t)\big) -L(t)u(t)+\nabla W(t,u(t))=0,\cr u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), }$$ where $\alpha\in (1/2,1)$, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive definite matrix. The novelty of this article is that if $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ and the nonlinearity $W(t,u)$ involves a combination of superquadratic and subquadratic terms, the Hamiltonian system possesses at least two nontrivial solutions.

Keywords

• Fractional Hamiltonian systems
• critical point
• variational methods
• mountain pass theorem