Global solution branches for \(p\)-Laplacian boundary value problems
type: Item
instance of: scholarly article
title: Global solution branches for \(p\)-Laplacian boundary value problems
zbMATH Open document ID: 1130.35065
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-06-27T00:00:00Z
review text: The authors study global continua of positive solutions to the Dirichlet problem \[ \begin{cases} -\Delta_p u= \lambda (1+u^q) & \text{in}\;\Omega\subset \mathbb R^n,\\ u=0 & \text{on}\;\partial\Omega. \end{cases} \] It is shown that, for subcritical values of \(q,\) there exists an unbounded continuum of positive solutions such that for every \(\lambda\in(0,\lambda^*)\) there are at least two solutions but for \(\lambda>\lambda^*\) no solution exists. Contrary, uniqueness holds for small positive \(\lambda\) when \(q\) is supercritical. The multiplicity result is proved by means of topological degree arguments combined with recent a~priori bounds, while the uniqueness for supercritical \(q\) follows from the Pokhozhaev identity and new weighted Poincaré-type inequalities.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 2183183
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: topological degree
zbMATH Keywords: a priori estimates
zbMATH Keywords: multiplicity
zbMATH Keywords: uniqueness
author: Jacqueline Fleckinger
author: W. Reichel
MaRDI profile type: Publication
full work available at URL: j.na.2003.11.015
OpenAlex ID: W2049106934
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DOI: 10.1016/J.NA.2003.11.015
description: scientific article; zbMATH DE number 2183183