Positive Ground State Solutions for Schrödinger-Poisson System with General Nonlinearity and Critical Exponent

Abstract

In this paper, we consider the following Schrödinger-Poisson system \begin{equation*}\begin{cases} -\Delta u + \eta\phi u = f(x,u) + u^5,& x\in\Omega,\\ -\Delta\phi=u^2,& x\in\Omega,\\u = \phi =0,& x\in \partial\Omega, \end{cases}\end{equation*} where $\Omega$ is a smooth bounded domain in $R^3$, $\eta=\pm1$ and the continuous function $f$ satisfies some suitable conditions. Based on the Mountain pass theorem, we prove the existence of positive ground state solutions.