Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials

TL;DR

This paper develops new geometric estimates for solutions to semilinear elliptic equations with singular, unbounded, and sign-changing potentials, bridging classical free boundary problems and extending regularity results.

Contribution

It introduces sharp regularity and nondegeneracy estimates for unbounded, sign-changing sources, expanding the understanding of elliptic problems with complex potentials.

Findings

Extended regularity estimates to unbounded, sign-changing sources

Provided nondegeneracy results for solutions with singular potentials

Connected free boundary problems with new geometric insights

Abstract

In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity) and obstacle-type. While previous studies have focused on bounded and strictly positive sources, we extend sharp regularity and nondegeneracy estimates to the unbounded, sign-changing setting, providing a comprehensive analysis of how the underlying nonlinearity interacts with minimal integrability assumptions on the source.