Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities

Abstract

We investigate interior regularity for stable solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in an open set $\Omega\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior H\"older estimates for stable classical solutions in every dimension $n\le 9$, thereby fully resolving a question raised by Cabr\'e concerning interior regularity without the nonnegativity assumption on $f$. For $n\le 5$, we also give a short alternative proof of the corresponding H\"older regularity result due to Peng, Zhang, and Zhou. We further establish higher-integrability estimates for the gradient and $L^1$ bounds for the Hessian in all dimensions. For $n\ge 10$, we obtain BMO and Morrey estimates, extending related results previously established for nonnegative nonlinearities. When $f$ is convex, we extend the interior regularity estimates to stable weak solutions through an approximation theorem. As applications, we prove Liouville-type theorems for entire stable solutions of equations with sign-changing nonlinearities across all dimensions.