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

  • Daniele Castorina

    Università di Roma Tor Vergata, Italy
  • Manel Sanchón

    Universitat de Barcelona, Spain

Abstract

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. 2949–2975

DOI 10.4171/JEMS/576