Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole
We consider the supercritical problem , where Θ is a bounded smooth domain in ℝ N , N ≥ 3, p > 2*:= 2 N /( N − 2), and Θ ∈ is obtained by deleting the ∈ -neighborhood of some sphere which is embedded in Θ. We show that in some particular situations, for small enough ∈ > 0, this problem has a p...
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Published in | Journal d'analyse mathématique (Jerusalem) Vol. 126; no. 1; pp. 341 - 357 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Jerusalem
The Hebrew University Magnes Press
01.04.2015
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | We consider the supercritical problem
, where Θ is a bounded smooth domain in ℝ
N
,
N
≥ 3,
p
> 2*:= 2
N
/(
N
− 2), and Θ
∈
is obtained by deleting the
∈
-neighborhood of some sphere which is embedded in Θ. We show that in some particular situations, for small enough
∈
> 0, this problem has a positive solution {itv}{in{it\te}} and that this solution concentrates and blows up along the sphere as
∈
→ 0. Our approach is to reduce this problem by means of a Hopf map to a critical problem of the form
, in a punctured domain
of lower dimension. We show that if Ω is a bounded smooth domain in ℝ
n
,
n
≥ 3,
is positive, and
, then for small enough
∈
> 0, this problem has a positive solution
u
ε
which concentrates and blows up at
Ξ
0
as
∈
→ 0. |
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Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
ISSN: | 0021-7670 1565-8538 |
DOI: | 10.1007/s11854-015-0020-6 |