Counterexamples to some conjectures on the number of solutions of nonlinear equations
type: Item
instance of: scholarly article
title: Counterexamples to some conjectures on the number of solutions of nonlinear equations
zbMATH Open document ID: 0556.35001
DOI: 10.1007/BF01455568
author: E. N. Dancer
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164022
review text: In this paper, we construct 3 counterexamples. Firstly, we construct an example which shows that a theorem of \textit{D. C. Clark} [Indiana Univ. Math. J. 22, 65-74 (1972; Zbl 0228.58006)] for nonlinear second order elliptic partial differential equations depends essentially on an oddness assumption. More precisely, we construct nonlinearities which cross many eigenvalues but the equation has only 4 nontrivial solutions. Secondly, we construct a related example which shows that a slight generalization of the Lazer-McKenna conjecture [see \textit{A. C. Lazer} and \textit{P. J. McKenna}, Proc. R. Soc. Edinb., Sect. A 95, 275-283 (1983; Zbl 0533.35037)] is false. The generalization consists in replacing the first eigenfunction by a function close to the first eigenfunction. Once again, we find examples where the nonlinearity crosses many eigenvalues but there are only 4 solutions. Thirdly, we construct competing species type models for two species with diffusion in which there are a large but finite number of solutions whatever the stability properties of the ''trivial'' solutions. Moreover, we find the exact number of solutions and their stability. The essential idea in all the examples is to perturb from degenerate situations.
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 3886359
zbMATH Keywords: nonlinear second order elliptic partial differential equations
zbMATH Keywords: competing species type models
zbMATH Keywords: exact number of solutions
Wikidata QID: Q123278760
MaRDI profile type: Publication
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OpenAlex ID: W2130905675
description: scientific article; zbMATH DE number 3886359