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Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications

João Vitor da Silva, Disson dos Prazeres, Gleydson Ricarte, Ginaldo Sá

Abstract

In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$ \mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^αu_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^α$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models.

Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications

Abstract

In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the conjecture for the -Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the -Laplacian with regular coefficients: under suitable assumptions on the data, with possibly singular weight , which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models.

Paper Structure

This paper contains 11 sections, 28 theorems, 245 equations, 1 table.

Table of Contents

  1. Introduction
  2. State-of-the-Art and brief motivations of the model equation
  3. Matukuma and Batt– Faltenbacher–Horst type equations
  4. Quasilinear models and its regularity theories
  5. Quasilinear equations: recent trends in regularity theory
  6. Preliminaries and auxiliary results
  7. Proof of Theorem \ref{['sharp_regularity']}
  8. Proof of Theorem \ref{['Hessian_continuity']}
  9. Non-degeneracy of solutions: Proof of Theorem \ref{['NãoDeg']}
  10. Proofs of Liouville-type Theorems
  11. Final comments

Key Result

Theorem 1.2

Let $u \in W^{1,p}(B_1)$ be a bounded weak solution to pobst for $f(|x|, u) = f(|x|)$. Suppose further that the assumptions condestr and intdoscoef hold. Then, $u \in C^{1, \beta}_{loc}(B_1)$, where and $\alpha_{\mathrm{H}} \in (0, 1)$ comes from Def_alpha_Hom. Moreover, the following estimates hold:

Theorems & Definitions (55)

  • Example 1.1
  • Theorem 1.2: Sharp/improved regularity estimates
  • Example 1.3
  • Corollary 1.4
  • Corollary 1.5: Optimal estimates in $2$-$\mathrm{D}$
  • Remark 1.6
  • Theorem 1.7: Higher Regularity Estimates
  • Remark 1.8
  • Remark 1.9: Equations of Matukuma and Batt– Faltenbacher–Horst type
  • Corollary 1.10: Sharp Gradient Growth
  • ...and 45 more