Lane-Emden Problems on Convex Domains of mathbb S²

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 $0 \leq p < 1$ the unique positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly concave in $\Omega$, while for $1 < p \leq 3$ 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 $0 \leq p \leq 3,$ any positive solution has strictly convex superlevel sets and a unique nondegenerate maximum.