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

By X. Cabré, Xavier Ros-Oton. 2012

64 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/2384f4394ff93c15289aaccb01f28cf499b2e885

Citations in the corpus

• Sharp isoperimetric inequalities via the ABP method (2013, unknown): “tive integers, and H is the Euclidean norm, Theorem 1.3 follows from the classical isoperimetric inequality in RD; see [12] for more details. SHARP ISOPERIMETRIC INEQUALITIES VIA THE ABP METHOD 11 In [11] we needed to show a Sobolev inequality of the type (1.3) in R2 with the weight and the cone giv”
• Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ (2019, unknown): “In particular, partial positive answers to the conjecture above have been given (chronologically): • by Nedev, when n ≤ 3 [30]; • by Cabré and Capella when Ω = B1 and n ≤ 9 [10]; • by Cabré when n = 4 and Ω is convex [7] (see [9] for an alternative proof); • by Villegas when n = 4 [39]; • by Cabré a”