Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Delta u=up in Omega, u=0 on (symbol omitted)Omega, where Omega is a bounded domain of {R}n, n is greater than or equal to 3, and p>2. We describe new concentration phenomena, which arise as p approaches the limit +infinity a...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2006-04, Vol.26 (2), p.201-225
Main Authors: Molle, Riccardo, Passaseo, Donato
Format: Article
Language:English
Subjects:
Calculus
Calculus of variations
Exponents
Mathematical analysis
Mathematical problems
Mathematics
Nonlinearity
Partial differential equations
Symmetry
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Summary:The paper deals with the existence of positive solutions of the problem -Delta u=up in Omega, u=0 on (symbol omitted)Omega, where Omega is a bounded domain of {R}n, n is greater than or equal to 3, and p>2. We describe new concentration phenomena, which arise as p approaches the limit +infinity and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p approached the limit +ininity, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Omega as p approaches the limit +infinity. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Omega is starshaped and p is greater than or equal to 2n/n-2, as a consequence of the Pohozaev's identity). [PUBLICATION ABSTRACT]
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-005-0364-3