Singular quasilinear elliptic equations and Hölder regularity
type: Item
instance of: scholarly article
title: Singular quasilinear elliptic equations and Hölder regularity
zbMATH Open document ID: 1275.35106
author: Jacques Giacomoni
author: Ian Schindler
author: Peter Takáč
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2012-06-04T00:00:00Z
reviewed by: Gary M. Lieberman
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J75
zbMATH DE Number: 6042967
zbMATH Keywords: singular elliptic equations
zbMATH Keywords: quasilinear elliptic equations
zbMATH Keywords: Hölder continuity
MaRDI profile type: Publication
full work available at URL: j.crma.2012.04.007
OpenAlex ID: W1972423504
cites work: Q3140494
cites work: C1 + α local regularity of weak solutions of degenerate elliptic equations
cites work: Q5428772
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Q5110819
cites work: Regularity of an elliptic problem with a singular nonlinearity
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: Regularity for a more general class of quasilinear equations
cites work: A strong maximum principle for some quasilinear elliptic equations
DOI: 10.1016/J.CRMA.2012.04.007
review text: The authors prove that solutions of the boundary value problem NEWLINE\[NEWLINE -\Delta_ pu = K(x)u^{-\delta}+g(x) \text{ in } \Omega, \;u=0 \text{ on } \partial\Omega NEWLINE\]NEWLINE are Hölder continuous when \(\Omega\) is an open, bounded domain in \(\mathbb R^ n\) with sufficiently smooth boundary, \(K\) satisfies the inequality \(0\leq K(x)\leq Cd(x)^{-\omega}\), \(g\) is nonnegative and bounded, and the positive constants \(\delta\) and \(\omega\) satisfy the inequality \(\omega<1+(1-\delta)(1-1/p)\). In addition, the solution \(u\) is assumed to be positive in \(\Omega\). When \(\omega+\delta<1\), the authors showed in a previous work [Ann. Sci. Norm. Sup. Pisa 6, 117--158 (2007; Zbl 1181.35116)] that \(u\) actually has Hölder continuous gradient, and the present paper is concerned with the case \(1\leq\omega+\delta\).NEWLINENEWLINE There is one peculiarity in the authors' Theorem 2.1: the statement of that theorem requires the exponent \(\beta\) to be less than the parameter \(\alpha\) but this restriction is not present in the proof of the theorem.
description: scientific article; zbMATH DE number 6042967