  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.
