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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Published in | Calculus of variations and partial differential equations Vol. 48; no. 3-4; pp. 611 - 623 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2013
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 0944-2669 1432-0835 |
DOI: | 10.1007/s00526-012-0564-6 |