Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents
We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N},\) \(N\geq3,\) and \(p\geq2^{*}:= 2N/(N-2).\) Bahri and Coron showed that if \(\Omega\) has nontrivial homology this problem has a positive s...
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Published in | arXiv.org |
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Main Authors | , , |
Format | Paper Journal Article |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
20.12.2012
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Subjects | |
Online Access | Get full text |
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Summary: | We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N},\) \(N\geq3,\) and \(p\geq2^{*}:= 2N/(N-2).\) Bahri and Coron showed that if \(\Omega\) 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 \(p\geq 2(N-1)/(N-3)\) 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 \(p> 2(N-k)/(N-k-2)\) with \(1\leq k\leq N-3\) there are bounded smooth domains in \(\mathbb{R}^{N}\) 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. |
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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1212.5137 |