Bifurcation diagrams of a \(p\)-Laplacian Dirichlet problem with Allee effect and an application to a diffusive logistic equation with predation
type: Item
instance of: scholarly article
title: Bifurcation diagrams of a \(p\)-Laplacian Dirichlet problem with Allee effect and an application to a diffusive logistic equation with predation
zbMATH Open document ID: 1213.34039
author: Kuo-Chih Hung
author: Shin-Hwa Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-01-12T00:00:00Z
review text: The authors study bifurcation diagrams of positive solutions for the \(p\)-Laplacian Dirichlet problem \[ (\varphi_p(u'(x)))'+\lambda f(u)=0,\quad u(-1)=u(1)=0, \] where \(p>1\), \(\varphi_p(y) = |y|^{p-2} y,\) \(\lambda> 0\) is a bifurcation parameter, \(f(y)=|y|^{p-1}g(y)\), and \(g\) is of Allee effect type. Assuming a suitable condition on \(g\), they prove that, on the (\(\lambda\), \(\|u\|_\infty\))-plane, the bifurcation diagram consists of exactly one continuous curve with exactly one turning point where the curve turns to the right. Hence, the problem has at most two positive solutions for each \(\lambda>0\). More precisely, they prove the exact multiplicity of positive solutions. Also, they give an application to a \(p\)-Laplacian diffusive logistic equation with predation of Holling type II functional response.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5835299
zbMATH Keywords: exact multiplicity
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: positive solution
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Allee effect
zbMATH Keywords: time map
Wikidata QID: Q115570420
reviewed by: Ruyun Ma
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.09.008
OpenAlex ID: W1981042221
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DOI: 10.1016/J.JMAA.2010.09.008
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description: scientific article; zbMATH DE number 5835299