On the structure of the positive solutions of the logistic equation with nonlinear diffusion
type: Item
instance of: scholarly article
title: On the structure of the positive solutions of the logistic equation with nonlinear diffusion
zbMATH Open document ID: 1042.35014
author: Manuel Delgado
author: Antonio Suárez
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/40184
review text: This paper is devoted to the structure of the positive solutions of the degenerate logistic equation, that is \[ \begin{cases} d{\mathcal J}w^m=\sigma w-b(x)w^z\quad &\text{in }\Omega\\ w=0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^N\), \(N\leq 1\), with a smooth boundary \(\partial\Omega\); \({\mathcal J}\) is a general second-order uniformly elliptic operator; \(b\) is a positive function, \(m\geq 1\), \(r>1\), \(d\) is a positive constant, and \(\sigma\) is a real parameter. An appropriate change of variable transforms (1) into \[ \begin{cases} {\mathcal J}u=\lambda u^q-b(x)u^p\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial\Omega,\end{cases}\tag{2} \] with \(\lambda\in \mathbb{R}\), \(0<q<p\), and \(q\leq 1\). Under suitable asumptions, on the data of (2) the authors prove that, there exists a unique positive solution of (2) if, and only if, \(\lambda>0\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1763453
zbMATH Keywords: positive solutions
zbMATH Keywords: degenerate logistic equation
zbMATH Keywords: uniqueness
zbMATH Keywords: nonlinear diffusion
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
OpenAlex ID: W2008404048
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DOI: 10.1006/JMAA.2001.7815
description: scientific article; zbMATH DE number 1763453