The extremal solution for the fractional Laplacian
We study the extremal solution for the problem ( - Δ ) s u = λ f ( u ) in Ω , u ≡ 0 in R n ∖ Ω , where λ > 0 is a parameter and s ∈ ( 0 , 1 ) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities w...
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| Veröffentlicht in: | Calculus of variations and partial differential equations Jg. 50; H. 3-4; S. 723 - 750 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article Verlag |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2014
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0944-2669, 1432-0835 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We study the extremal solution for the problem
(
-
Δ
)
s
u
=
λ
f
(
u
)
in
Ω
,
u
≡
0
in
R
n
∖
Ω
, where
λ
>
0
is a parameter and
s
∈
(
0
,
1
)
. We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions
n
<
4
s
. We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever
n
<
10
s
. In the limit
s
↑
1
,
n
<
10
is optimal. In addition, we show that the extremal solution is
H
s
(
R
n
)
in any dimension whenever the domain is convex. To obtain some of these results we need
L
q
estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with
L
p
data. We prove optimal
L
q
and
C
β
estimates, depending on the value of
p
. These estimates follow from classical embedding results for the Riesz potential in
R
n
. Finally, to prove the
H
s
regularity of the extremal solution we need an
L
∞
estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels. |
|---|---|
| Bibliographie: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0944-2669 1432-0835 |
| DOI: | 10.1007/s00526-013-0653-1 |