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An elliptic operator with a nonlocal term: maximum principle, principal eigenvalue, and applications to a logistic equation with indefinite weight

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  24 February 2025

Willian Cintra Open the ORCID record for Willian Cintra [Opens in a new window]  and
Ismael Oliveira dos Anjos Open the ORCID record for Ismael Oliveira dos Anjos [Opens in a new window]
Willian Cintra*
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil (willian@unb.br) (corresponding author)
Ismael Oliveira dos Anjos
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil (ismaeloliveira.mat@gmail.com)
*
Corresponding author: Cintra Willian, email: willian@unb.br
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Abstract

In this work, we consider a class of uniformly elliptic operators with a nonlocal term and mixed boundary conditions in bounded domains. We establish the existence of a principal eigenvalue and provide a result that offers both sufficient and necessary conditions for the validity of the maximum principle. As a consequence of these findings, we conduct a detailed study of an eigenvalue problem with an indefinite weight, as well as establish existence and uniqueness results for a logistic-type equation and prove some blow-up results.

Keywords

logistic equationmaximum principlenonlocal elliptic operatorprincipal eigenvalue

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35B09: Positive solutions 35B50: Maximum principles 92D25: Population dynamics (general)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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