Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

By Xavier Cabré Vilagut, A. Kort. 2005

115 citations recorded by the source corpus.

We consider a special class of radial solutions of semilinear equations −?u = g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution u ? H1 0 is bounded if n ? 9 (for every g), and belongs to H3 = W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/982d237eba2c505a3aea8344ae0e0d75fd81e143

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Citations in the corpus

Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ (2019, background): “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”
Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result (2008, background): “Related geometric inequalities also played an important role in [6].”
The extremal solution for the fractional Laplacian (2013, background): “In the unit ball Ω = B1, it is well known that the extremal solution for s = 1 is bounded whenever n ≤ 9 and may be singular if n ≥ 10 [7], while the extremal solution for s = 2 is bounded whenever n ≤ 12 and may be singular if n ≥ 13 [13].”
Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations (2010, background): “[6] Cabré, X., Capella, A. (2006).”
Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian (2006, background): “On the other hand, the authors and Capella [10] extend the radial results of [9] for p = 2 to the case p > 1 obtaining, for instance, the boundedness of every radial semi-stable solution when N < p+ 4p/(p− 1).”
Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution (2012, methodology): “The proofs of the results in [2, 3, 6, 15, 16] use the semi-stability of the extremal solution u∗.”