Logistic Dirichlet problems with discontinuous coefficients.
type: Item
instance of: scholarly article
title: Logistic Dirichlet problems with discontinuous coefficients.
zbMATH Open document ID: 1064.35059
DOI: 10.1016/S0021-7824(03)00058-8
author: Kazuaki Taira
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-12-03T00:00:00Z
review text: The author studies existence of positive solutions to semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. Precisely, let \(\Omega\subset {\mathbb R}^N,\) \(N\geq3,\) be a bounded and \(C^{1,1}\)-smooth domain and \[ {\mathcal L}:= -\sum_{i,j=1}^N a^{ij}(x){{\partial^2}\over{\partial x_i\partial x_j}} +\sum_{i=1}^N b^{i}(x){{\partial}\over{\partial x_i}} +c(x) \] a uniformly elliptic operator with \(a^{ij}\in VMO\cap L^\infty,\) \(b^i,\;c\in L^\infty,\) \(c(x)\geq0\) for almost all \(x\in\Omega.\) The paper deals with the problem \[ \begin{cases} {\mathcal L}u=\lambda\big(m(x)u-h(x)u^2\big) &\text{in}\;\Omega,\\ u=0 &\text{on}\;\partial\Omega, \end{cases} \] where \(\lambda>0\) is a parameter, \(m\in C(\overline\Omega)\) may change its sign in \(\Omega\) whereas \(h\in C(\overline\Omega)\) and \(h(x)>0\) in \(\overline\Omega.\) A complete study is carried out of the changes which occur in the global structure of positive solutions to the above Dirichlet problem as the parameter \(\lambda\) varies from the principal eigenvalue \(\lambda_1(m)\) of the linearized problem \[ \begin{cases} {\mathcal L}u=\lambda m(x)u &\text{in}\;\Omega,\\ u=0 &\text{on}\;\partial\Omega. \end{cases} \]
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 2012734
zbMATH Keywords: diffusive logistic equation
zbMATH Keywords: indefinite weight function
zbMATH Keywords: Dirichlet boundary condition
zbMATH Keywords: uniformly elliptic operator
zbMATH Keywords: VMO coefficients
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2012734