Abstract:

In this paper, we consider the following critical Schrödinger-Poisson system \begin{eqnarray*} \left\{ {\begin{array}{*{20}{l}}{\begin{array}{*{20}{l}}{ - \Delta u + \lambda V{\rm{(}}x{\rm{)}}u + \phi u = \mu |u{|^{p - 2}}u + |u{|^4}u{\rm{, }}\; \; \; }\\{ - \Delta \phi = {u^2}, \; \; \; \; \; \; \; }\end{array}\begin{array}{*{20}{c}}{x \in {\mathbb{R}^3},}\\{x \in {\mathbb{R}^3},}\end{array}}\end{array}} \right. \end{eqnarray*} 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 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 solutions if $\lambda$ is large and $\mu$ is small.

Key words: Critical exponent, Asymptotical behavior, Multiple solutions