Regularity of stable solutions up to dimension 7 in domains of double revolution

2012/02/06 by Xavier Cabré, Xavier Cabre, Xavier Ros-Oton +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1202.1220

arxiv created 2012/02/06 · openalex publication_date 2012/02/06 · arxiv updated 2012/02/07 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider the class of semi-stable positive solutions to semilinear equations -Δu=f(u) in a bounded domain Ω⊂\mathbb Rⁿ 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 H¹₀ 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.

Citations

• A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations

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