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Archive for Rational Mechanics and Analysis

, Volume 207, Issue 3, pp 1075–1089 | Cite as

A Concentration Phenomenon for Semilinear Elliptic Equations

  • Nils Ackermann
  • Andrzej Szulkin
Article

Abstract

For a domain \({\Omega \subset \mathbb{R}^{N}}\) we consider the equation
$$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$
with zero Dirichlet boundary conditions and \({p\in(2, 2^*)}\). Here \({V \geqq 0}\) and Q n are bounded functions that are positive in a region contained in \({\Omega}\) and negative outside, and such that the sets {Q n  > 0} shrink to a point \({x_0 \in \Omega}\) as \({n \to \infty}\). We show that if u n is a nontrivial solution corresponding to Q n , then the sequence (u n ) concentrates at x 0 with respect to the H 1 and certain L q -norms. We also show that if the sets {Q n  > 0} shrink to two points and u n are ground state solutions, then they concentrate at one of these points.

Keywords

Soliton Nontrivial Solution Dielectric Response Kerr Nonlinearity Ground State Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Circuito Exterior, C.U.MexicoMexico
  2. 2.Department of MathematicsStockholm UniversityStockholmSweden

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