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Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian

Elzon C. Bezerra Júnior, João Vitor da Silva, Thialita M. Nascimento, Ginaldo S. Sá

Abstract

In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hardy-Hénon-type equations with possibly singular weights and strong absorption governed by the $\infty$-Laplacian $$ Δ_{\infty} u(x) = |x|^αu_+^m(x) \quad \text{in} \quad B_1, $$ 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.

Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian

Abstract

In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hardy-Hénon-type equations with possibly singular weights and strong absorption governed by the -Laplacian 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 -Laplacian. As an application of our findings, we also address some Liouville-type results for this class of equations.

Paper Structure

This paper contains 13 sections, 21 theorems, 165 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Further results and implications
  4. The Borderline Scenario: A Strong Maximum Principle
  5. A relevant review of the literature
  6. Hardy-Hénon-type models
  7. $\infty$-Laplacian and its Existence/Regularity Theories
  8. Elliptic diffusion models related to our research
  9. Preliminaries results
  10. Proof of Theorem \ref{['Hessian_continuity']}
  11. Non-degeneracy estimate: Proof of Theorem \ref{['NãoDeg']}
  12. Proof of Liouville-type Theorem \ref{['Liouville I']}
  13. Radial analysis and proof of Theorem \ref{['Liouville II']}

Key Result

Theorem 1

Let $u \in C^0(B_1)$ be a viscosity solution of problem pobst. Assume that $f(|x|, t) \simeq |x - x_0|^{\alpha} t_+^m$ with EqHomog-f in force. Then, for any point $x_0 \in B_{1/2} \cap \partial \{ u > 0 \}$, we have for $r \in (0, 1/2)$, where $\mathrm{C} > 0$ is a universal constantThroughout this manuscript, a universal constant is the one depending only on universal parameters, namely $m$, $\

Theorems & Definitions (37)

  • Example 1: A toy model
  • Theorem 1: Higher regularity estimates
  • Remark 1.1
  • Remark 1.2
  • Corollary 1: Gradient growth
  • Theorem 2: Non-degeneracy at critical points
  • Example 2: A one-dimensional profile
  • Remark 1.3
  • Example 3
  • Example 4: A radial profile
  • ...and 27 more