Global bifurcation results for semilinear elliptic equations on \(\mathbb{R}^N\): The Fredholm case
type: Item
instance of: scholarly article
title: Global bifurcation results for semilinear elliptic equations on \(\mathbb{R}^N\): The Fredholm case
zbMATH Open document ID: 0899.35009
published in: Journal of Differential Equations
publication date: 1998-11-17T00:00:00Z
review text: The nonlinear eigenvalue problem \[ -\Delta u=\lambda g(x)f(u)\quad \text{in } \mathbb{R}^N\;(N\geq 3),\;0<u<1,\;\lim_{| x|\to 0} u(x)=0 \] is studied. Here \(g\) is an \(L^\infty\) smooth function such that \(g(x)>0\) on \(\Omega^+\) with \(| \Omega^+ |>0\) and there exist \(k\), \(R_0>0\) such that \(g(x) <-k\) if \(| x| >R_0\), and \(f:[0,1] \to\mathbb{R}^+\) is a smooth function such that \(f(0)=f(1) =0\), \(f'(0) >0\) \(f'(1) <0\) and \(f(u) >0\) if \(u\in (0,1)\), satisfying additional regularity conditions. The main result states that under these assumptions, and if \(N= 3,4,5\), a global bifurcation theorem of Rabinowitz's type holds, giving the existence of a branch of positive solutions such that \(0<u<1\). In Section 2 the problem is settled in the framework of Sobolev spaces and an extension of Poincaré's inequality is proved. The associated eigenvalue problem is considered in Section 3 and the existence of a principal eigenvalue with a positive eigenfunction in \(H^2\) is shown. The existence of a local branch of positive solutions is proved in Section 4 by using Crandall-Rabinowitz's theorem. The difficulty raised by the noncompactness of the Fredholm operators involved is overcomed in Section 5 by using an extension of degree theory due to Dancer and an important result concerning the compactness of the solution set is given in Section 6. Finally, Section 7 is devoted to the main bifurcation theorem and some complementary results.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 1130545
zbMATH Keywords: global bifurcation
zbMATH Keywords: branch of positive solutions
zbMATH Keywords: Poincaré's inequality
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: degree theory
author: Nikos M. Stavrakakis
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3346
OpenAlex ID: W2056567796
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DOI: 10.1006/JDEQ.1997.3346
description: scientific article; zbMATH DE number 1130545