Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

Abstract

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla u{|}^{\alpha }{\Delta }_p^{N}u=1$ has the property that $u^{\frac{\alpha +1}{\alpha +2}}$ is a concave function. Secondly we consider positive solutions of the eigenvalue problem $-|\nabla u{|}^{\alpha }{\Delta }_p^{N}u=\lambda |u{|}^{\alpha }u$, in which case $\log u$ turns out to be concave. The methods provided include a weak comparison principle and a Hopf-type Lemma.