On the uniqueness of positive solution of an elliptic equation
type: Item
instance of: scholarly article
title: On the uniqueness of positive solution of an elliptic equation
zbMATH Open document ID: 1125.35037
published in: Applied Mathematics Letters
publication date: 2006-01-24T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/40112
review text: The authors consider the following semilinear elliptic boundary value problem: \[ \begin{cases} -\Delta u = f(x,u) & \text{ in }\Omega,\\ u = 0 & \text{ on }\partial \Omega, \end{cases}\tag{1} \] with \(\Omega\) regular domain in \(\mathbb R^N\) and \(f : \Omega \times \mathbb R \to \mathbb R\) measurable function. The main result is the following: Assume that there exists \(g \in C([0, +\infty))\cap C^1(0, +\infty)\), such that \(g(t) > 0\) for \(t > 0\), \(g'\) is non-increasing and the map \(u \to f(x,u)/g(u)\) is non-increasing in \((0, +\infty)\) for almost every \(x \in \Omega\). Then problem (1) has a unique nonnegative solution. It is shown also that \(u(x) > 0\) \(\forall x \in \Omega\) and \(\partial u/\partial n < 0\) on \(\partial \Omega\).
reviewed by: Davide Guidetti
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5001416
zbMATH Keywords: uniqueness of positive solutions
author: Manuel Delgado
author: Antonio Suárez
MaRDI profile type: Publication
OpenAlex ID: W2034965973
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DOI: 10.1016/J.AML.2004.09.020
description: scientific article; zbMATH DE number 5001416