Necessary and sufficient conditions of existence for a system involving the \(p\)-Laplacian (\(1 < p < N\))
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions of existence for a system involving the \(p\)-Laplacian (\(1 < p < N\))
zbMATH Open document ID: 1016.35020
DOI: 10.1016/S0022-0396(02)00094-3
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The existence of positive solutions for the system \[ -\Delta_pu=\mu f(x,u,v)\quad\text{in } \mathbb{R}^N,\quad -\Delta_pv=\mu g(x,u,v)\quad\text{in } \mathbb{R}^N,\quad \lim_{|x|\to\infty}u (x)=0=\lim_{|x|\to\infty}v (x), \] where \(\Delta_p\) is the \(p\)-Laplacian \((1<p<N)\) is studied. For \(0<\mu\leq \mu_*\) there exists a positive solution and for \(\mu> \mu^*\) the system has no positive solution. In some case \(\mu_*=\mu^*\) because these parameters depend on \(p\), \(N\) and the growth of \(f\) and \(g\). The existence result is related to subsolution and supersolution methods and the nonexistence result is a consequence of Picone's identity.
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1915524
zbMATH Keywords: positive solution
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Picone identity
zbMATH Keywords: subsolution
zbMATH Keywords: supersolution
author: Karim Chaib
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915524