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A global result for a degenerate quasilinear eigenvalue problem with discontinuous nonlinearities

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Abstract

This paper considers a class of degenerate quasilinear elliptic equations with discontinuous nonlinearities. The existence of positive weak solutions and S-solutions is discussed using variational methods. The results assert that the \((\lambda ,a)\)-space of the parameters involved is divided into three regions - no solution, at least one S-solution, and at least two weak solutions (one is S-solution among them), in each region respectively. The regions are separated by a continuous, nondecreasing curve and line segment. Further, there exists an S-solution at each point on the separating curve.

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Acknowledgements

The authors thank the referee for careful reading and for many useful comments. The first author would like to thank Ailton Rodrigues da Silva (DMAT/UFRN) for several discussions about this subject.

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. J. Abrantes Santos was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq) – 303479/2019-1.

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Author notes

  1. J. A. Santos, P. F. S. Pontes, S. H. M. Soares have contributed equally to this work.

Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Av. Aprígio Veloso, 785, Campina Grande, Paraíba, 58109-970, Brazil

    Jefferson Abrantes Santos & Pedro F. Silva Pontes

  2. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trabalhador São-carlense, 400, São Carlos, São Paulo, 13566-590, Brazil

    Sergio H. Monari Soares

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  1. Jefferson Abrantes Santos
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  2. Pedro F. Silva Pontes
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Correspondence to Sergio H. Monari Soares.

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Santos, J.A., Pontes, P.F.S. & Soares, S.H.M. A global result for a degenerate quasilinear eigenvalue problem with discontinuous nonlinearities. Calc. Var. 62, 91 (2023). https://doi.org/10.1007/s00526-023-02437-2

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