References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt147",{id:"formSmash:upper:j_idt147",widgetVar:"widget_formSmash_upper_j_idt147",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt148_j_idt150",{id:"formSmash:upper:j_idt148:j_idt150",widgetVar:"widget_formSmash_upper_j_idt148_j_idt150",target:"formSmash:upper:j_idt148:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

A concentration phenomenon for semilinear elliptic equationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 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]

##### Place, publisher, year, edition, pages

2013. Vol. 207, no 3, 1075-1089 p.
##### Keyword [en]

Concentration, semilinear elliptic equation
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-83444DOI: 10.1007/s00205-012-0589-1ISI: 000314026900012OAI: oai:DiVA.org:su-83444DiVA: diva2:575786
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt388",{id:"formSmash:j_idt388",widgetVar:"widget_formSmash_j_idt388",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt394",{id:"formSmash:j_idt394",widgetVar:"widget_formSmash_j_idt394",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt400",{id:"formSmash:j_idt400",widgetVar:"widget_formSmash_j_idt400",multiple:true});
##### Funder

Swedish Research Council
Available from: 2012-12-11 Created: 2012-12-11 Last updated: 2013-03-18Bibliographically approved

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.

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