一类带有两个参数的临界薛定谔-泊松方程的多重解

Abstract

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

CLC Number:

O175.2

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

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