#### Mohsen Timoumi, Infinitely many solutions for two classes of fractional Hamiltonian systems, Vol. 2021 (2021), Article ID 6, pp. 1-17

Full Text: PDF
DOI: 10.23952/cot.2021.6

Received February 24, 2021; Accepted May 11, 2021; Published May 25, 2021

Abstract. In this paper, we study the existence and multiplicity of solutions for a class of fractional Hamiltonian systems
${l} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R},$
$u\in H^{\alpha}(\mathbb{R}),$
where $L(t)$ satisfies some weaker conditions than the well-known conditions, and $W(t,x)$ is of superquadratic growth as $\left|x\right|\longrightarrow\infty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|x|\longrightarrow\infty$). To the best of our knowledge, there is no result concerning the existence and multiplicity of solutions for the above system with our conditions. The proof is based on variational methods and critical point theory.