The effect of domain shape on the number of positive solutions of certain nonlinear equations
type: Item
instance of: scholarly article
title: The effect of domain shape on the number of positive solutions of certain nonlinear equations
zbMATH Open document ID: 0662.34025
DOI: 10.1016/0022-0396(88)90021-6
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The author considers how the shape of the bounded domain \(\Omega\) \((\Omega \in R^ m\), \(m>1)\) affects the number of positive solution to the problem (i) \(-\Delta u=\lambda f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(f\in C^ 1[{\mathbb{R}},{\mathbb{R}}]\). For instance, he proves that if \(f(u)=\exp u\), then there are contractible domains \(\Omega\) for which equation (i) has large number of solutions. The case \(f(u)=u^ p\) is also under particular attention. The conditions assuring that the problem (i) (with \(f(u)=u^ p)\) has a unique positive, nondegenerate solution are given. A lot of examples illustrate the obtained results.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4081997
zbMATH Keywords: bifurcation point
zbMATH Keywords: starshaped domain
zbMATH Keywords: positive solution
zbMATH Keywords: examples
author: E. N. Dancer
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90021-6
OpenAlex ID: W4212869559
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description: scientific article; zbMATH DE number 4081997