Regularity of stable solutions up to dimension 7 in domains of double revolution
Abstract
We consider the class of semi-stable positive solutions to semilinear equations
−Δu=f(u)
in a bounded domain
Ω⊂
R
n
of double revolution, that is, a domain invariant under rotations of the first
m
variables and of the last
n−m
variables. We assume
2≤m≤n−2
. When the domain is convex, we establish a priori
L
p
and
H
1
0
bounds for each dimension
n
, with
p=∞
when
n≤7
. These estimates lead to the boundedness of the extremal solution of
−Δu=λf(u)
in every convex domain of double revolution when
n≤7
. The boundedness of extremal solutions is known when
n≤3
for any domain
Ω
, in dimension
n=4
when the domain is convex, and in dimensions
5≤n≤9
in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities
f
in dimensions
5≤n≤9
.