Stable solutions to semilinear elliptic equations for operators with variable coefficients

TL;DR

This paper extends interior regularity results for stable solutions of semilinear elliptic equations with variable coefficients, establishing Hölder continuity in dimensions up to 9 with bounds depending on lower regularity norms.

Contribution

The work generalizes previous regularity results to operators with variable coefficients, reducing regularity assumptions needed for bounds on stable solutions.

Findings

Stable solutions are Hölder continuous in dimensions up to 9.

Bounds depend on the $C^1$ norm of coefficients, not higher derivatives.

Results are independent of the specific nonlinearity $f$, as long as it is non-negative.

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.