In this paper we look for positive solutions of the problem -Delta u + lambda u = u(p-1) in Omega, u = 0 on a partial derivative Omega, where Omega is a bounded domain in R-n, n >= 3, p > 2 and lambda is a positive parameter. We describe new concentration phenomena, which occur as lambda --> +infinity, and exploit them to construct (for lambda large enough) positive solutions that concentrate near spheres of codimension 2 as lambda --> +infinity; these spheres approach the boundary of Omega as lambda --> +infinity. Notice that the existence and multiplicity results we obtain hold also in contractible domains arbitrarily close to starshaped domains (no solution can exist if p >= 2n/n-2 and Omega is starshaped, because of Pohozaev's identity). The method we use is completely variational and based on a blow up analysis in the equivariant setting. In order to avoid concentration phenomena near points and to overcome some difficulties related to the lack of compactness, we first modify the nonlinear term in a suitable region, then we solve the modified problem by minimizing the related energy functional on a suitable infinite dimensional manifold and, finally, we show that the solutions of the modified problem solve also our problem, for lambda large enough, because they are localized in the prescribed region where the nonlinear term has not been modified.
Molle, R., & Passaseo, D. (2006). Concentration phenomena for solutions of superlinear elliptic problems. ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE, 23(1), 63-84 [10.1016/j.anihpc.2005.02.002].
|Tipologia:||Articolo su rivista|
|Citazione:||Molle, R., & Passaseo, D. (2006). Concentration phenomena for solutions of superlinear elliptic problems. ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE, 23(1), 63-84 [10.1016/j.anihpc.2005.02.002].|
|IF:||Con Impact Factor ISI|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Revisione (peer review):||Sì, ma tipo non specificato|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.anihpc.2005.02.002|
|Stato di pubblicazione:||Pubblicato|
|Data di pubblicazione:||2006|
|Titolo:||Concentration phenomena for solutions of superlinear elliptic problems|
|Autori:||Molle, R ; Passaseo, D|
|Appare nelle tipologie:||01 - Articolo su rivista|