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On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  28 December 2022

Marcos T. O. Pimenta Open the ORCID record for Marcos T. O. Pimenta [Opens in a new window] ,
Gelson Conceição G. dos Santos  and
João R. Santos Júnior
Marcos T. O. Pimenta
Affiliation:
Departamento de Matemática e Computação, Universidade Estadual Paulista - Unesp, CEP: 19060-900, Presidente Prudente - SP, Brazil (marcos.pimenta@unesp.br)
Gelson Conceição G. dos Santos
Affiliation:
Faculdade de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém - PA, Brazil (gelsonsantos@ufpa.br, joaojunior@ufpa.br)
João R. Santos Júnior
Affiliation:
Faculdade de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém - PA, Brazil (gelsonsantos@ufpa.br, joaojunior@ufpa.br)
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Abstract

In this work, we study a quasilinear elliptic problem involving the 1-Laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - \beta )$. Our approach is based on an analysis of the associated p-Laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $\beta \to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

Keywords

1-Laplacian operatorspace of functions of bounded variationdiscontinuous nonlinearities

MSC classification

Primary: 35J62: Quasilinear elliptic equations
Secondary: 35J75: Singular elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 33 - 59
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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