Existence and nonexistence of solutions for quasilinear elliptic equations
type: Item
instance of: scholarly article
title: Existence and nonexistence of solutions for quasilinear elliptic equations
zbMATH Open document ID: 0956.35055
author: Marcelo Montenegro
author: Marcos Montenegro
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-02-28T00:00:00Z
review text: The paper deals with the Dirichlet problem: \[ -\Delta_p u=\text{div}\left(|\nabla u|^{p-2}\nabla u\right)= f(x,u,\nabla u)\quad\text{in}\quad\Omega, u=0\quad \text{on }\partial\Omega, \] where \(\Omega\) is a bounded \(C^2\) domain in \(\mathbb{R}^N\), \(N\geq 2\), \(1<p<\infty\) and the nonlinearity \(f(x,t,\xi)\) satisfies the following critical growth condition \[ a(t) |\xi|^p+b(x,t)\leq f(x,t,\xi)\leq c(t) |\xi|^p+d(x,t). \] There are obtained some necessary and sufficient conditions on continuous functions \(a, c: \mathbb{R}^1_+\to \mathbb{R}^1_+\) and Caratheodory functions \(b,d:\Omega\times\mathbb{R}^1_+\to \mathbb{R}^1_+\) under which the problem under consideration admits a weak solution.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1471616
zbMATH Keywords: quasilinear elliptic degenerate equation
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: critical growth in the gradient
zbMATH Keywords: existence and nonexistence of solution
reviewed by: A. E. Shishkov
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6697
OpenAlex ID: W2066278829
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DOI: 10.1006/JMAA.1999.6697
description: scientific article; zbMATH DE number 1471616