The branch of positive solutions to a semilinear elliptic equation on \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: The branch of positive solutions to a semilinear elliptic equation on \(\mathbb{R}^N\)
zbMATH Open document ID: 0933.35060
author: H. Jeanjean
author: Marcello Lucia
author: C. A. Stuart
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2000-03-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108488
review text: Summary: For a large class of functions \(f\), we consider the nonlinear elliptic eigenvalue problem \[ \begin{aligned} -\Delta u(x)+f\bigl(x,u(x)\bigr)=\lambda u(x)\quad & \text{for }x \in\mathbb{R}^N,\\ \lim_{|x|\to\infty} u(x)=0, \quad u(x)>0\quad & \text{for all }x\in \mathbb{R}^N.\end{aligned} \] If the lowest eigenvalue of the linearization lies below the essential spectrum, it is known [*, \textit{H. Jeaujean}, \textit{M. Lucia} and \textit{C. A. Stuart}, Math. Z. 230, No. 1, 79-105 (1999)] that a global branch, \({\mathcal C}\), of solutions bifurcates at this value of \(\lambda\). The main results of the present paper give properties of \(f\) which enable us to provide explicit lower bounds for \(\sup\{\lambda\mid(\lambda,u)\in{\mathcal C}\}\). Our approach is based on the construction of supersolutions which enable us to refine the alternative established in [*] concerning the global behaviour of \({\mathcal C}\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1319604
zbMATH Keywords: supersolutions
zbMATH Keywords: explicit bounds
zbMATH Keywords: global behaviour
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1319604