Regularity of minimizers of semilinear elliptic problems up to dimension 4

By X. Cabré. 2009

113 citations recorded by the source corpus.

We consider the class of semistable solutions to semilinear equations −Δu = f(u) in a bounded smooth domain Ω of \input amssym $\Bbb R^n$ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an a priori L∞‐bound that holds for every positive semistable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n = 4 and Ω is convex. This result was previously known only in dimensions n ≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω = BR by a result of A. Capella and the author. © 2010 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1775007be9b5e889d60806df98a95152927d23b8

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Citations in the corpus

Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ (2019, methodology): “This was crucially used in [7], in combination with the Sobolev-type inequality of Michael-Simons and Allard, to prove regularity of stable solutions up to dimension n ≤ 4.”
The extremal solution for the fractional Laplacian (2013, background): “Villegas [36] have proved, using the results in [6], the boundedness of the extremal solution in dimension n = 4 for all domains, not necessarily convex.”
Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution (2012, methodology): “1 in B 1 ˆRnis given by u= 1 2n (1 s2 t) and, thus, u s= 1 n s is only Lipschitz in . Before proving Proposition 1.6, we will need two preliminary results. The rst one, Lemma 2.2, was already used in [3, 2]. In this paper we use it taking the function con its statement to be u s and u t. Note that c”
Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations (2010, methodology): “This result has been extended by Cabré to the case n = 4 and strictly convex [5].”