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Positive solutions for a degenerate Kirchhoff problem

Published online by Cambridge University Press:  19 August 2021

David Arcoya
Affiliation:
Dpto. de Análisis Matemático, Univ. de Granada, Granada18071, Spain ([email protected])
João R. Santos Júnior
Affiliation:
Faculdade de Matemática, Instituto de Ciências Exatas e Naturais, Univ. Federal do Pará, Avda. Augusto Corrêa 01, Belém, PA66075-110, Brazil ([email protected])
Antonio Suárez
Affiliation:
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Sevilla, Spain ([email protected])

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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