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Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

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Abstract

In this paper we address the following Kirchhoff type problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta (g(|\nabla u|_2^2) u + u^r) = a u + b u^p&{} \quad \text{ in }~\varOmega , \\ u>0&{}\quad \text{ in }~\varOmega , \\ u= 0&{} \quad \text{ on }~\partial \varOmega , \end{array} \right. \end{aligned}$$

in a bounded and smooth domain \(\varOmega \) in \(\mathbb {R}^{N}\). By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters a, b, p, r and the non-linearity g, the existence of positive solutions.

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Acknowledgements

The authors thank to the referee for her/his comments and suggestions which improve notably this work.

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Authors and Affiliations

  1. Faculdade de Matemática, Instituto de Ciências Exatas e Naturais, Universidade Federal do Pará, Avenida Augusto corrêa 01, Belém, PA, 66075-110, Brazil

    Willian Cintra & João R. Santos Júnior

  2. Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-090, Brazil

    Gaetano Siciliano

  3. Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, Calle Tarfia s/n, Sevilla, Spain

    Antonio Suárez

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  1. Willian Cintra
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  2. João R. Santos Júnior
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  3. Gaetano Siciliano
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  4. Antonio Suárez
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Correspondence to Willian Cintra.

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W. Cintra was partially supported by CAPES Proc. No BEX 6377/15-7, Brazil. J. R. Santos Jr. was partially supported by CNPq-Proc. 302698/2015-9 and CAPES-Proc. 88881.120045/2016-01, Brazil. A. Suárez has been partially supported by MTM2015-69875-P (MINECO/FEDER, UE) and CNPq-Proc. 400426/2013-7. G. Siciliano was partially supported by Fapesp, Capes and CNPq, Brazil.

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Cintra, W., Santos Júnior, J.R., Siciliano, G. et al. Existence results of positive solutions for Kirchhoff type equations via bifurcation methods. Math. Z. 295, 1143–1161 (2020). https://doi.org/10.1007/s00209-019-02385-8

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  • DOI: https://doi.org/10.1007/s00209-019-02385-8

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