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Positive solutions for a degenerate Kirchhoff problem

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 August 2021

David Arcoya ,
João R. Santos Júnior  and
Antonio Suárez
David Arcoya
Affiliation:
Dpto. de Análisis Matemático, Univ. de Granada, Granada 18071, Spain (darcoya@ugr.es)
João R. Santos Júnior
Affiliation:
Faculdade de Matemática, Instituto de Ciências Exatas e Naturais, Univ. Federal do Pará, Avda. Augusto Corrêa 01, Belém, PA 66075-110, Brazil (joaojunior@ufpa.br)
Antonio Suárez
Affiliation:
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Sevilla, Spain (suarez@us.es)
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Abstract

By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$-norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

Keywords

degenerate Kirchhoff problemvariational methodsconcentration phenomena

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35Q74: PDEs in connection with mechanics of deformable solids

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 675 - 688
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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