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

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.