Improved regularity estimates for Hardy-H\'{e}non-type equations driven by the $\infty$-Laplacian

TL;DR

This paper establishes sharp regularity estimates for solutions of Hardy-Hénon-type equations involving the ∞-Laplacian, extending previous results and providing new geometric and Liouville-type insights.

Contribution

It introduces explicit regularity exponents for solutions with singular weights and strong absorption, improving and extending prior regularity results for ∞-Laplacian driven equations.

Findings

Derived explicit regularity exponents depending on universal parameters.

Proved non-degeneracy properties of solutions.

Extended results to zero-obstacle and dead-core problems.

Abstract

In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hardy-H\'{e}non-type equations with possibly singular weights and strong absorption governed by the $\infty$-Laplacian $$ \Delta_{\infty} u(x) = |x|^{\alpha}u_+^m(x) \quad \text{in} \quad B_1, $$ under suitable assumptions on the data. In this setting, we derive an explicit regularity exponent that depends only on universal parameters. Additionally, we prove non-degeneracy properties, providing further geometric insights into the nature of these solutions. Our regularity estimates not only improve but also extend, to some extent, the previously obtained results for zero-obstacle and dead-core problems driven by the $\infty$-Laplacian. As an application of our findings, we also address some Liouville-type results for this class of equations.