Regularity of minimizers of semilinear elliptic problems up to dimension 4

Abstract

We consider the class of semistable solutions to semilinear equations-Δu=f(u)bounded smooth domain Ω of R{double-struck}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.