Regularity of Minimizers of Semilinear Elliptic Problems up to Dimension Four

Abstract. We consider the class of semi-stable solutions to semilinear equations −∆u = f(u) in a bounded smooth domain Ω of R (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an priori L∞ bound which holds for every semi-stable 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.