Multiple solutions for a class of quasilinear problems
type: Item
instance of: scholarly article
title: Multiple solutions for a class of quasilinear problems
zbMATH Open document ID: 1161.35366
author: Francisco Odair de Paiva
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-04-10T00:00:00Z
review text: The author deals with the existence of positive and multiple solutions for the quasilinear elliptic problem \[ \begin{aligned} -\Delta_pu= g(x,u)\quad &\text{in }\Omega,\tag{1}\\ u= 0\quad &\text{on }\partial\Omega,\tag{2}\end{aligned} \] where \(\Omega\subset\mathbb{R}^d\) is an open bounded domain with smooth boundary \(\partial\Omega\) and \(g: \Omega\times\mathbb{R}\to \mathbb{R}\) is a Carathéodory function such that \(g(x,0)= 0\), which implies that (1)--(2) possesses the trivial solution \(u= 0\). The author is interested in nontrivial solutions. Here \(\Delta_p\) denotes the \(p\)-Laplace operators that is, \(\Delta_p u:= \text{div}(|\nabla u|^{p-2}\nabla u)\). Under some natural assumptions on \(g(x,u)\) he establishes the existence of positive and multiple solutions for (1)--(2). To this end the author uses a variational approach.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5141170
zbMATH Keywords: quasilinear problems
zbMATH Keywords: indefinite weight
zbMATH Keywords: multiplicity of solutions
MaRDI profile type: Publication
full work available at URL: dcds.2006.15.669
OpenAlex ID: W2092591214
DOI: 10.3934/DCDS.2006.15.669
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description: scientific article; zbMATH DE number 5141170