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

TL;DR

This paper investigates a quasilinear elliptic problem with the 1-Laplacian operator and a discontinuous nonlinearity, analyzing the asymptotic behavior of solutions as parameters approach critical limits.

Contribution

It introduces a novel analysis of the 1-Laplacian problem with discontinuous nonlinearity via the p-Laplacian approach and studies solution limits as parameters tend to zero.

Findings

Solutions converge as p approaches 1+

Solutions tend to a limit as β approaches 0+

The limit solves the original discontinuous problem

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.