Regularity of stable solutions to semilinear elliptic equations on Riemannian models

Abstract

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian manifolds includes the classical \textit{space forms}, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish $L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend on the nonlinearity $f$. As an application, under standard assumptions on the nonlinearity $\lambda f(u)$, we prove that the corresponding extremal solution $u^*$ is bounded whenever $n\leq9$. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.