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

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    • Keywords


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

    • Abstract


      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.

    • Author Affiliations



      1. Depatamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Peru
    • Dates

  • Proceedings – Mathematical Sciences | News

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