Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition
type: Item
instance of: scholarly article
title: Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition
zbMATH Open document ID: 1235.35127
author: K. Umezu
published in: Journal of Differential Equations
publication date: 2011-12-14T00:00:00Z
review text: This paper is devoted to the study of positive solutions of the following semilinear elliptic problem with a nonlinear boundary condition \[ \begin{cases} -\Delta u=\lambda\left(m(x)u-u^2\right) &\text{ in }\Omega,\\ \frac{\partial u}{\partial \mathbf{n}}=\lambda b(x)v^2 &\text{ on }\partial\Omega, \end{cases} \tag{1} \] where \(\Omega\subset \mathbb R^N\), \(N=2, 3\), is a bounded domain with a smooth boundary \(\partial\Omega\). \(m\in C^{\theta}(\Omega)\) and \(b\in C^{\theta+1}(\Omega)\) are Hölder continuous functions with exponent \(0 <\theta<1\), and \(m\) is positive somewhere in \(\Omega\), that is, it may change sign, while \(b>0\) on \(\partial\Omega\). The author separately considers the cases that \(\int_{\Omega}mdx<0\), \(\int_{\partial\Omega}b\phi_1^3ds\geq\int_{\Omega}\phi_1^3dx\) and \(\int_{\Omega}mdx\geq0\). In the case of \(\int_{\Omega}mdx<0\) and \(\int_{\partial\Omega}b\phi_1^3ds\geq\int_{\Omega}\phi_1^3dx\), the positive solution that bifurcates subcritically at \((\lambda_1, 0)\) is unique and unstable for any \(\lambda\) less than but close to \(\lambda_1\). Additionally, if \(\int_{\partial\Omega}bds\leq |\Omega|\), the positive solution \((\lambda, u_\lambda)\) satisfies \(\lim_{\lambda\downarrow0}\min_{x\in \overline{\Omega}}u_\lambda(x)\to\infty\) and if \(\int_{\partial\Omega}bds> |\Omega|\), \(u_\lambda\to C(m,b)\) in \(C^{2+\theta}(\overline{\Omega})\) as \(\lambda\downarrow0\), \(C(m,b)\in \mathbb R\) dependents on functions \(m,b\). Then in the case of \(\int_{\Omega}mdx\geq0\), if \(\int_{\partial\Omega}bds< |\Omega|\), the first solution bifurcating from \(\{(0,c):c\geq 0 \text{ is a constant}\}\) is unstable and it tends to \(\infty\) in \(C(\overline{\Omega})\) as \(\lambda\downarrow0\), the second one is unstable for \(\lambda\) sufficiently small and its minimum value in \(\overline{\Omega}\) tends to infinity as \(\lambda\downarrow0\) and if \(\int_{\partial\Omega}bds\geq |\Omega|\), problem (1) has no positive solution for …
reviewed by: Chun-Lei Tang
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 5988422
zbMATH Keywords: semilinear elliptic problem
zbMATH Keywords: indefinite weight
zbMATH Keywords: positive solution
zbMATH Keywords: global subcontinuum
zbMATH Keywords: linearized stability
zbMATH Keywords: Lyapunov and Schmidt procedure
zbMATH Keywords: scaling argument
zbMATH Keywords: population dynamics
MaRDI profile type: Publication
full work available at URL: j.jde.2011.08.043
OpenAlex ID: W1989636246
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DOI: 10.1016/J.JDE.2011.08.043
description: scientific article; zbMATH DE number 5988422