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space of functions of bounded variation

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity
Part of
- Elliptic equations and systems
Marcos T. O. Pimenta, Gelson Conceição G. dos Santos, João R. Santos Júnior
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 1 / February 2024

Published online by Cambridge University Press:

28 December 2022, pp. 33-59

Print publication:

February 2024
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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.