The Nehari manifold for a semilinear elliptic equation involving a sublinear term
type: Item
instance of: scholarly article
title: The Nehari manifold for a semilinear elliptic equation involving a sublinear term
zbMATH Open document ID: 1130.35049
published in: Calculus of Variations and Partial Differential Equations
publication date: 2005-04-25T00:00:00Z
review text: The author discusses the problem of existence and multiplicity of non-negative solutions to the problem \[ \begin{cases} -\Delta u(x)=\lambda u(x)+b(x)| u(x)| ^{\gamma-2}u(x)& \text{for }u\in\Omega\\ u(x)=0& \text{for }x\in\partial\Omega,\end{cases}\eqno(1) \] where \(\Omega\subset \mathbb R^N\) is a smooth bounded domain, \(b:\Omega\to \mathbb R\) a smooth function, \(\lambda\in \mathbb R\) and \(1<\gamma<2\). When \(1<\gamma<2\) the problem \((1)\) is asymptotically linear and the author establishes results on bifurcation from infinity when \(\lambda=\lambda_{1}\), the principal eigenvalue of the linear problem \[ -\Delta u(x)=\lambda u(x)\quad x\in\Omega;\qquad u(x)=0\quad x\in\partial\Omega. \] By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form \(t\mapsto J(tu)\) where \(J\) is the Euler functional associated to \((1)\)), the author studies how the Nehari manifold changes as \(\lambda\) varies. The bifurcation is then described in terms of the sign of the quantity \(\int_{\Omega}b\phi_{1}^{\gamma}\,dx\) where \(\phi_{1}\) is the positive eigenfunction of the above linear problem corresponding to \(\lambda_{1}\).
reviewed by: Piero Montecchiari
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2161207
zbMATH Keywords: Nehari manifold
zbMATH Keywords: fibering maps
zbMATH Keywords: semilinear sublinear elliptic equations
author: Kenneth J. Brown
MaRDI profile type: Publication
full work available at URL: s00526-004-0289-2
OpenAlex ID: W2056249562
Wikidata QID: Q115387811
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DOI: 10.1007/S00526-004-0289-2
description: scientific article; zbMATH DE number 2161207