Regularity of stable solutions of pp-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 Δpu=g(u)–\Delta_p u= g(u) in a smooth bounded domain ΩRn\Omega\subset \mathbb{R}^n. In particular, we obtain new LrL^r and W1,rW^{1,r} bounds for the extremal solution uu^\star when the domain is strictly convex. More precisely, we prove that uL(Ω)u^\star\in L^{\infty}(\Omega) if np+2n\leq p+2 and uLnpnp2(Ω)W01,p(Ω)u^\star \in L^{\frac{np}{n-p-2}} (\Omega) \cap W^{1,p}_0 (\Omega) if n>p+2n > p + 2.

Cite this article

Daniele Castorina, Manel Sanchón, Regularity of stable solutions of pp-Laplace equations through geometric Sobolev type inequalities. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2949–2975

DOI 10.4171/JEMS/576