Archive for Rational Mechanics and Analysis

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

A Concentration Phenomenon for Semilinear Elliptic Equations



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 Qn are bounded functions that are positive in a region contained in \({\Omega}\) and negative outside, and such that the sets {Qn > 0} shrink to a point \({x_0 \in \Omega}\) as \({n \to \infty}\). We show that if un is a nontrivial solution corresponding to Qn, then the sequence (un) concentrates at x0 with respect to the H1 and certain Lq-norms. We also show that if the sets {Qn > 0} shrink to two points and un are ground state solutions, then they concentrate at one of these points.


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© 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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