一类带有两个参数的临界薛定谔-泊松方程的多重解

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1750-1767.

一类带有两个参数的临界薛定谔-泊松方程的多重解

陈永鹏¹(),杨志鹏^2,*()

¹ 广西科技大学理学院广西柳州 545006
² Georg-August-University of Göttingen, Göttingen 37073

收稿日期:2020-09-11 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 杨志鹏 E-mail:yongpengchen@mail.bnu.edu.cn;yangzhipeng326@163.com
作者简介:陈永鹏, E-mail: yongpengchen@mail.bnu.edu.cn
基金资助:
广西高校中青年教师科研基础能力提升项目(2017KY1383);广西高校中青年教师科研基础能力提升项目(2021KY0348)

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

Yongpeng Chen¹(),Zhipeng Yang^2,*()

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)

摘要/Abstract

摘要：

该文研究如下一类临界薛定谔-泊松方程\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*}其中$\lambda>0，\mu>0$是两个参数，$p\in（4，6）$，$V$满足一些势井条件.当参数$\lambda$充分大时，利用变分法证明了基态解的存在性，以及随着$\lambda\to\infty$时，这些解的渐近行为.另外，在参数$\lambda$充分大和$\mu$充分小时，利用Ljusternik-Schnirelmann理论，到了多重解的存在性定理.

关键词: 临界指标, 渐近行为, 多重解

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

中图分类号:

O175.2

陈永鹏,杨志鹏. 一类带有两个参数的临界薛定谔-泊松方程的多重解[J]. 数学物理学报, 2021, 41(6): 1750-1767.

Yongpeng Chen,Zhipeng Yang. Multiplicity of Solutions for a Class of Critical Schrödinger-Poisson System with Two Parameters[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1750-1767.

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