Lane-Emden Problems on Convex Domains of ${\mathbb{S}}^2$

Abstract

We study positive solutions of the Dirichlet problem $-\Delta u=u^p$ in a uniformly convex domain $\Omega \subset {\mathbb{S}}^2$, $u=0$ on $\partial \Omega .$ For $p=1$, we assume that the right-hand side is replaced by ${\lambda }_{1}u$, where ${\lambda }_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with zero Dirichlet boundary condition. We prove that for $0\le p<1$ the unique positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly concave in $\Omega$, while for $1<p\le 3$ every positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly convex in $\Omega .$ For $p=0,$ our result gives the strict $1/2-$concavity of the torsion function in $\Omega .$ For $p=1,$ a result due to Lee and Wang gives the strict log-concavity of the first eigenfunction in $\Omega .$ As a consequence, for each $0\le p\le 3,$ any positive solution has strictly convex superlevel sets and a unique nondegenerate maximum.