Multiplicity of solutions for a class of critical Schrödinger-Poisson system with two parameters

Acta Math. Sci. Ser. A (Chinese Ed.) 41 (2021), no. 6, 1750–1767

The following critical Schr"odinger-Poisson system is considered: \begin{equation*} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u, \end{equation*} \begin{equation*} -\Delta \phi=u^2, \end{equation*} where $\lambda, \mu$ are two positive parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of positive ground state solutions for $\lambda$ large enough and $\mu>0$, and their asymptotical behavior as $\lambda\to\infty$. Moreover, by using Lusternik-Schnirelmann theory, we obtain the existence of multiple positive solutions if $\lambda$ is large and $\mu$ is small.