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January, 2011 Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential
David Ruiz , Giusi Vaira
Rev. Mat. Iberoamericana 27(1): 253-271 (January, 2011).

Abstract

In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrödinger Equation.

Citation

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David Ruiz . Giusi Vaira . "Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential." Rev. Mat. Iberoamericana 27 (1) 253 - 271, January, 2011.

Information

Published: January, 2011
First available in Project Euclid: 4 February 2011

zbMATH: 1216.35024
MathSciNet: MR2815737

Subjects:
Primary: 35B40 , 35J20 , 35J55

Keywords: multi-bump solutions , nonlinear analysis , Schrödinger-Poisson-Slater problem , singular perturbation method , variational methods

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.27 • No. 1 • January, 2011
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