Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system

Yanqin Fang; Jihui Zhang

doi:10.3934/cpaa.2011.10.1267

Article Contents

2011, Volume 10, Issue 4: 1267-1279. Doi: 10.3934/cpaa.2011.10.1267

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Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system

Yanqin Fang^1, and
Jihui Zhang^1,

1.
Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu

Received: March 31, 2010

Revised: October 31, 2010

Published: June 2011

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Abstract

In this paper, we study the following system

$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J10, 35B35.

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References

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