Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

In this paper, we consider the following Schrödinger-Poisson equation\begin{document}$ \left\{\begin{aligned} &-\triangle u + u + \phi u = u^{5}+\lambda g(u), &\hbox{in}\ \ \Omega, \\\ & -\triangle \phi = u^{2}, & \hbox{in}\ \ \Omega, \\\ & u, \phi = 0, & \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $\end{document}where \begin{document}$ \Omega $\end{document} is a bounded smooth domain in \begin{document}$ \mathbb{R}^{3} $\end{document} , \begin{document}$ \lambda>0 $\end{document} and the nonlinear growth of \begin{document}$ u^{5} $\end{document} reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.