Regularity of minimizers of semilinear elliptic problems up to dimension four

TL;DR

This paper proves that semi-stable solutions to semilinear elliptic equations are uniformly bounded in dimensions up to four, extending previous results and highlighting open questions in higher dimensions.

Contribution

It establishes a priori $L^ abla$ bounds for semi-stable solutions in dimensions up to four, including extremal solutions in convex domains, generalizing earlier work.

Findings

Boundedness of semi-stable solutions in $n \\leq 4$

Extension of boundedness results to extremal solutions in convex domains

Open problem remains for dimensions $5 \\leq n \\leq 9$

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 $n \leq 4$, 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 $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n \leq 9$ 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.