Stable solutions to semilinear elliptic equations for operators with variable coefficients

Abstract

In this paper we extend the interior regularity results for stable solutions in [Cabr\'{e}, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation $a_{ij}(x)u_{ij} + b_i(x) u_i + f(u) = 0$ are H\"{o}lder continuous in the optimal range of dimensions $n \leq 9$. Our bounds are independent of the nonlinearity $f \in C^1$, which we assume to be non-negative. The main achievement of our work is to make the constants in our estimates depend on the $C^1$ norm of $a_{ij}$ and the $C^0$ norm of $b_i$, instead of their $C^2$ and $C^1$ norms, respectively, which arise in a first approach to the computations.