\(W^{1,p}\) versus \(C^1\) local minimizers and multiplicity results for quasilinear elliptic equations.
type: Item
instance of: scholarly article
title: \(W^{1,p}\) versus \(C^1\) local minimizers and multiplicity results for quasilinear elliptic equations.
zbMATH Open document ID: 1160.35382
DOI: 10.1016/S0022-247X(03)00282-8
author: Zongming Guo
author: Zhitao Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-11-16T00:00:00Z
review text: The paper considers the existence and multiplicity of positive (negative) solutions for the equation \[ \begin{gathered} \text{div}(|\nabla u(x)|^{p-2}\nabla u(x))+ \lambda| u(x)|^{q-1} u(x)+ g(u(x))= 0\quad\text{in }\Omega,\\ u\in W^{1,p}_0(\Omega),\end{gathered}\tag{1} \] where \(\Omega\subset\mathbb{R}^n\), \(n\geq 3\), is a bounded smooth domain, \(p> 1\), \(0< q< p=1\), \(g(u)u\geq 0\) and \(g\) has a growth larger than or equal to \(p-2\). Most results are obtained for \(p> 2\) and a typical one is that there exists \(\Lambda> 0\) such that for \(\lambda>\Lambda\) the equation (1) has no positive solutions, but for \(0<\lambda<\Lambda\) there are at least two positive solutions.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2003886
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: multiplicity of solutions
reviewed by: Uldis Raitums
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00282-8
OpenAlex ID: W2030084221
description: scientific article; zbMATH DE number 2003886