The existence of three positive solutions of a singular \(p\)-Laplacian problem
type: Item
instance of: scholarly article
title: The existence of three positive solutions of a singular \(p\)-Laplacian problem
zbMATH Open document ID: 1250.35102
author: Lin Zhao
author: Yuan He
author: Peihao Zhao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-07-22T00:00:00Z
review text: In this paper, the authors deal with the existence and multiplicity of solutions to the following boundary value problem: \[ -\Delta_pu = \lambda a(x)u^{-\gamma}+\lambda f(x,u),\,\,\,x\in\Omega, \;u>0\,\,\,\text{in}\,\,\,\Omega, \;u=0\,\,\, \text{on}\,\,\,\partial\Omega, \tag{1} \] where \(\Omega\) is a bounded smooth open domain of \(\mathbb R^N\), \(\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian, \(N<p<+\infty\), \(\gamma>0\) is a constant, \(\lambda>0\) is a parameter, \(f\) is a Carathéodory function on \(\Omega\times[0,+\infty)\), and \(a(x)\geqslant 0\) is a nontrivial measurable function satisfying the condition: there are \(\varphi_0\geqslant 0\) in \(C_0^1(\overline\Omega)\) and \(q>N\) such that \(a\varphi_0^{-\gamma}\in L^q(\Omega)\). Under some additional assumptions on the function \(f\), the authors establish that there exist an open interval \(\Sigma\subset (0,+\infty)\) and a constant \(\mu>0\) such that for every \(\lambda\in\Sigma\) problem (1) has at least three distinct positive solutions in \(W_0^{1,p}(\Omega)\), with their \(W_0^{1,p}(\Omega)\) norms less than \(\mu\).
reviewed by: Alexander A. Kovalevsky
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35J92
zbMATH DE Number: 5930967
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: singular nonlinearity
zbMATH Keywords: positive solutions
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.na.2011.05.065
OpenAlex ID: W2076431193
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DOI: 10.1016/J.NA.2011.05.065
description: scientific article; zbMATH DE number 5930967