Multiplicity of Positive Solutions for Schrödinger–Poisson Systems with a Critical Nonlinearity in \(\mathbb {R}^3\)

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Abstract

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrödinger–Poisson problem
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\phi u=\lambda Q(x)|u|^{q-2}u+ K(x)|u|^4u, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ -\Delta \phi =u^2, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ \end{array}\right. \end{aligned}$$
where \(4<q<6 \) or \(q=2\), \(\lambda >0\) is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small \(\lambda \).

Keywords

Multiple positive solutions Schrödinger–Poisson systems Variational methods Barycenter map Critical Sobolev exponent 

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.School of Mathematics and Computational ScienceHunan First Normal UniversityChangshaChina

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