Regularity of extremal solutions of semilinear elliptic equations with m-convex nonlinearities

TL;DR

This paper generalizes the boundedness results of extremal solutions for semilinear elliptic equations with m-convex nonlinearities, extending previous findings and providing a unified framework through stability and blow-up analysis.

Contribution

It extends the boundedness results of extremal solutions to general m-convex nonlinearities and offers a unified perspective using stability and blow-up techniques.

Findings

Extends boundedness results to m-convex nonlinearities.

Provides a closedness result for stable solutions.

Uses blow-up arguments to prove extremal solution boundedness.

Abstract

We consider the Gelfand problem in a bounded smooth domain $\Omega\subset \mathbb{R}^N$ with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution $u^*$. When the dimension $N\ge10$, it is known that a singular extremal solution can be constructed for the nonlinearity $f(u)=e^u$ and $\Omega=B_1$. When $3\le N\le 9$, Cabr\'e, Figalli, Ros-Oton, and Serra (2020) proved the following surprising result: the extremal solution $u^*$ is bounded if the nonlinearity $f$ is positive, nondecreasing, and convex. In this paper, we succeed in generalizing their result to general $m$-convex nonlinearities. Moreover, we give a unified viewpoint on the results of previous studies by considering $m$-convexity. We provide a closedness result for stable solutions with $m$-convex nonlinearities. As a consequence, we provide a Liouville-type result and by using a blow-up argument, we prove the boundedness of extremal solutions.