Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (6): 1750-1767.

### Multiplicity of Solutions for a Class of Critical Schrödinger-Poisson System with Two Parameters

Yongpeng Chen1(),Zhipeng Yang2,*()

1. 1 School of Science, Guangxi University of Science and Technology, Guangxi Liuzhou 545006
2 Mathematical Institute, Georg-August-University of Göttingen, Göttingen 37073
• Received:2020-09-11 Online:2021-12-26 Published:2021-12-02
• Contact: Zhipeng Yang E-mail:yongpengchen@mail.bnu.edu.cn;yangzhipeng326@163.com
• Supported by:
the Basic Ability Improvement Project of Young and Middle-Aged Teachers in Guangxi Universities(2017KY1383);the Basic Ability Improvement Project of Young and Middle-Aged Teachers in Guangxi Universities(2021KY0348)

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.

CLC Number:

• O175.2
Trendmd