Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent

Abstract

In this article we consider the fractional Schrodinger-Poisson system $$\displaylines{ (-\Delta)^{s} u - \mu \frac{\Phi(x/|x|)}{|x|^{2s}} u +\lambda \phi u = |u|^{2^*_s-2}u,\quad \text{in } \mathbb{R}^3,\cr (-\Delta)^t \phi = u^2, \quad \text{in } \mathbb{R}^3, }$$ where $s\in(0,3/4)$, $t\in(0,1)$, $2t+4s=3$, $\lambda>0$ and $2^*_s=6/(3-2s)$ is the Sobolev critical exponent. By using perturbation method, we establish the existence of a solution for $\lambda$ small enough.

Keywords

• fractional schrodinger-poisson system
• weighted hardy potential
• critical exponent