Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition
type: Item
instance of: scholarly article
title: Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition
zbMATH Open document ID: 1257.34013
author: Guowei Dai
author: Yanqiong Lu
author: Ruyun Ma
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-10-29T00:00:00Z
review text: Using some topological methods and nonlinear analytic techniques, the authors prove a global bifurcation result from infinity for the \(p\)-Laplacian problem \[ -(\varphi_p(u'))'=\lambda a(t)\varphi_p(u)+g(t,u;\lambda),\,\,\,t\in (0,1);\,\,\,u(0)=u(1)=0, \] where \(\varphi_p(s)=|s|^{p-2}s\), and \(a:[0,1]\to [0,+\infty)\) and \(f:\mathbb{R}\to\mathbb{R}\) are continuous functions, which satisfy some additional assumptions. As an application, they investigate the global behavior of the components of nodal solutions of the nonlinear eigenvalue problem \[ (\varphi_p(u'))'+\lambda a(t)f(u)=0,\,\,\,t\in (0,1);\,\,\,u(0)=u(1)=0. \] The intervals for the parameter \(\lambda\) which ensure the existence of multiple nodal solutions if \(f_0\in (0,\infty)\) and \(f_{\infty}\in (0,\infty)\) are also established, where \(f_0=\lim_{s\to 0}f(s)/\varphi_p(s)\) and \(f_{\infty}=\lim_{|s|\to\infty}f(s)/\varphi_p(s)\).
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6099585
zbMATH Keywords: \(p\)-Laplacian problem
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: bifurcation
zbMATH Keywords: nodal solutions
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.07.056
OpenAlex ID: W2070877027
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DOI: 10.1016/J.JMAA.2012.07.056
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description: scientific article; zbMATH DE number 6099585