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

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(middot - 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 -> 1(+). We study also the asymptotic behaviour of the solutions, as beta -> 0(+) and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.