A Positive Solution of a Schrödinger-Poisson System with Critical Exponent

Abstract

We use variational methods to study the existence of at least one positive solution of the following Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} \Delta u +u +l(x)\phi u = k(x)|u|^{{2^*}2}u +\mu h(x)|u|^{q2}u \quad & \ \hbox{in}\ \mathbb{R}^3,\\ \\ \Delta \phi = l(x)u^2\quad & \ \hbox{in} \ \mathbb{R}^3, \end{array} \right. $$ under some suitable conditions on the nonnegative functions $l, k, h$ and constant $\mu\gt 0$, where $2\leq q\lt 2^*$ (critical Sobolev exponent).