Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem where Ω is a bounded smooth domain in , N  ≥ 3, and . Bahri and Coron showed that if Ω has nontrivial homology this problem has a positive solution for p  = 2 * . However, this is not enough to guarantee existence in the supercritical case. For Passaseo exhibite...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 48; no. 3-4; pp. 611 - 623
Main Authors Clapp, Mónica, Faya, Jorge, Pistoia, Angela
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2013
Springer Nature B.V
Subjects
Analysis
Calculus of variations
Calculus of Variations and Optimal Control; Optimization
Control
Mathematical and Computational Physics
Mathematical problems
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
35J20
35J60
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Summary:We consider the supercritical problem where Ω is a bounded smooth domain in , N  ≥ 3, and . Bahri and Coron showed that if Ω has nontrivial homology this problem has a positive solution for p  = 2 * . However, this is not enough to guarantee existence in the supercritical case. For Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as p increases. More precisely, we show that for with 1 ≤ k  ≤ N −3 there are bounded smooth domains in whose cup-length is k  + 1 in which this problem does not have a nontrivial solution. For N  = 4,8,16 we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-012-0564-6