Harnack inequality for non-divergence structure semi-linear elliptic equations
type: Item
instance of: scholarly article
title: Harnack inequality for non-divergence structure semi-linear elliptic equations
zbMATH Open document ID: 1400.35053
author: Ahmed Mohammed
author: Giovanni Porru
published in: Advances in Nonlinear Analysis
publication date: 2018-08-16T00:00:00Z
review text: In this paper, the authors prove a Harnack inequality for non-negative solutions of \(Lu = f(u)\), where \(L\) is a non-divergence structure uniformly elliptic operator and \(f\) is a non-decreasing function that satisfies an appropriate growth conditions at infinity. In [Trans. Am. Math. Soc. 363, No. 10, 5131--5178 (2011; Zbl 1225.35076)], \textit{M. Dindoš} proved a Harnack inequality for the solutions of the equation \(\Delta u = f(u)\), where \(f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) is non-decreasing and continuous such that \(f(t) > 0\) for \(t > 0\), and there is \(\theta > 1\) such that \(\underset{t\rightarrow \infty }\liminf \frac{f(\theta t)}{\theta f(t)} >1.\) In this paper, the authors extend Dindoš' Harnack inequality results to more general operators and provide an example to show that Dindoš' condition cannot be too much relaxed for the stated Harnack inequality to hold.
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 6919837
zbMATH Keywords: non-divergence structure elliptic operator
zbMATH Keywords: large solution
zbMATH Keywords: Alexandroff-Bakelman-Pucci maximum principle
MaRDI profile type: Publication
full work available at URL: anona-2016-0050
OpenAlex ID: W2434361286
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DOI: 10.1515/ANONA-2016-0050
description: scientific article; zbMATH DE number 6919837