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Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations

  • Xiaoming He
Article

Abstract

This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schrödinger–Poisson equations
$$ \left\{ \begin{array}{l@{\quad}l} -\varepsilon^2\triangle u+V(x)u+\phi(x) u=f(u)& {\rm in}\,{\mathbb R}^3, \\ -\varepsilon^2\triangle \phi=u^2 & {\rm in}\,{\mathbb R}^3, \\ u\in H^1({\mathbb R}^3), u(x) > 0,& \forall x\in{\mathbb R}^3, \\ \end{array} \right. $$
where ε > 0 is a parameter, \({V: {\mathbb R}^3\rightarrow{\mathbb R}}\) is a continuous function and \({f: {\mathbb R}\rightarrow {\mathbb R}}\) is a C 1 function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik–Schnirelmann theory.

Mathematics Subject Classification (2000)

35J60 35J25 

Keywords

Positive solutions Schrödinger–Poisson equation Variational methods 

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.College of ScienceMinzu University of ChinaBeijingPeople’s Republic of China

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