Regularity of minimizers of semilinear elliptic problems up to dimension four

Abstract

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n\le 4$, we establish an priori $L^{\infty }$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $\Omega$ is convex. This result was previously known only in dimensions $n\le 3$ by a result of G. Nedev. In dimensions $5\le n\le 9$ the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case $\Omega =B_R$ by a result of A. Capella and the author.