# Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

created by castorina on 22 Apr 2016
modified on 31 Oct 2018

[BibTeX]

Published Paper

Inserted: 22 apr 2016
Last Updated: 31 oct 2018

Journal: Journal of the European Mathematical Society
Year: 2015

ArXiv: 1201.3486 PDF
In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-\Delta_p u= g(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.