Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

Proceedings of the Indian Academy of Sciences. Mathematical sciences

CÉSAR E TORRES LEDESMA^1¹ 

[1] Depatamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Peru^1

关键词: Liouville–Weyl fractional derivative;    fractional Sobolev space;    critical point theory;    variational method;    positive semi-definite;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this article, we consider the following fractional Hamiltonian systems:$$_{t}D^{\alpha}_{\infty}(_{−\infty}D^{\alpha}_{t} u) + \lambda L(t)u = \Delta W(t, u), t \in \mathbb{R},$$where $\alpha \in (1/2, 1), \lambda$ > 0 is a parameter, $L \in C (\mathbb{R, R}^{n\times n})$ and $W \in C^{1}(\mathbb{R \times R}^{n}, \mathbb{R})$. Unlike most other papers on this problem, we require that $L(t)$ is a positive semi-definite symmetric matrix for all $t \in \mathbb{R}$, that is, $L(t) \equiv 0$ is allowed to occur in some finite interval $\mathbb{I}$ of $\mathbb{R}$. Under some mild assumptions on $W$, we establish the existence of nontrivial weak solution, which vanish on $\mathbb{R \backslash I}$ as $\lambda \rightarrow \infty$, and converge to $\tilde{u}$ in $H^{\infty}(\mathbb{R})$; here $\tilde{u} \in E^{\alpha}_{0}$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb{I}$. Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

【 授权许可】

CC BY