Concave power solutions of the Dominative $p$-Laplace equation

Abstract

In this paper, we study properties of solutions of the Dominative $p$-Laplace equation with homogeneous Dirichlet boundary conditions in a bounded convex domain $\Omega$. For the equation $-{\mathcal{D}}_{p}u=1$, we show that $\sqrt{u}$ is concave, and for the eigenvalue problem ${\mathcal{D}}_{p}u+\lambda u=0$, we show that $\log u$ is concave.