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 a priori estimates for semistable solutions of in a smooth bounded domain . In particular, we obtain new and bounds for the extremal solution when the domain is strictly convex. More precisely, we prove that if and if .
Cite this article
Daniele Castorina, Manel Sanchón, Regularity of stable solutions of -Laplace equations through geometric Sobolev type inequalities. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2949–2975DOI 10.4171/JEMS/576