On a fractional Schrödinger-Poisson system with strong singularity

Abstract

We investigate a fractional Schrödinger-Poisson system with strong singularity as follows: (−Δ)su+V(x)u+λϕu=f(x)u−γ,x∈R3,(−Δ)tϕ=u2,x∈R3,u>0,x∈R3,\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+V\left(x)u+\lambda \phi u=f\left(x){u}^{-\gamma },& x\in {{\mathbb{R}}}^{3},\\ {\left(-\Delta )}^{t}\phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\\ u\gt 0,& x\in {{\mathbb{R}}}^{3},\end{array}\right. where 034s+2t\gt 3, λ>0\lambda \gt 0 and γ>1\gamma \gt 1. When VV and ff satisfy certain conditions, existence and uniqueness of positive solution uλ{u}_{\lambda } are established via variational method and Nehari method. We also describe the asymptotic behaviour of uλ{u}_{\lambda } as λ→0\lambda \to 0.

Keywords

• strong singularity
• fractional schrödinger-poisson system
• uniqueness
• variational method
• asymptotic behaviour
• 35a15
• 35b40
• 35b09
• 35r11