Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution

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 e...

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Bibliographic Details
Published in:Communications in partial differential equations 2013-01, Vol.38 (1), p.135-154
Main Authors: Cabre, Xavier, Ros-Oton, Xavier
Format: Article
Language:English
Subjects:
Dimensional stability
Estimates
Invariants
Mathematical analysis
Nonlinearity
Partial differential equations
Regularity
Regularity of stable solutions
Semilinear elliptic equations
Studies
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Summary: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 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 ≤ 4 for any domain Ω, 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.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2012.697505