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 [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation aij(x)uij+bi(x)ui+f(u) = 0 are Hölder continuous in the optimal range of dimensions n ≤ 9. Our bounds are independent of the nonlinearity f ∈ C1, which we assume to be nonnegative. The main achievement of our work is to make the constants in our estimates depend on the C1 norm of aij and the C0 norm of bi, instead of their C2 and C1 norms, respectively, which arise in a first approach to the computations. © 2023 American Institute of Mathematical Sciences. All rights reserved.