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A concentration phenomenon for semilinear elliptic equations
Stockholm University, Faculty of Science, Department of Mathematics.
2013 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 207, no 3, 1075-1089 p.Article in journal (Refereed) Published
Abstract [en]

For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ 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.

Place, publisher, year, edition, pages
2013. Vol. 207, no 3, 1075-1089 p.
Keyword [en]
Concentration, semilinear elliptic equation
National Category
Research subject
URN: urn:nbn:se:su:diva-83444DOI: 10.1007/s00205-012-0589-1ISI: 000314026900012OAI: diva2:575786
Swedish Research Council
Available from: 2012-12-11 Created: 2012-12-11 Last updated: 2013-03-18Bibliographically approved

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Szulkin, Andrzej
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