Nontrivial solution for Schrödinger–Poisson equations involving the fractional Laplacian with critical exponent

Abstract

This paper deals with a class of nonlocal Schrodinger equations with critical exponent $$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s} u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=f(x,u),&{}\quad \mathrm{in}\ \mathbb {R}^3,\\ (-\Delta )^s \phi =K(x)|u|^{2^*_s-1},&{}\quad \mathrm{in}\ \mathbb {R}^3. \end{array}\right. \end{aligned}$$ By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of nontrivial solution is obtained under appropriate assumptions on V, K and f.