On the behaviour of the solutions to $p$-Laplacian equations as $p$ goes to $1$

In the present paper we study the behaviour as $p$ goes to $1$ of the weak
solutions to the problems
$$
\begin{cases}
-\operatorname{div} \bigl(|\nabla u_p|^{p-2}\nabla u_p\bigr)=f
&\text{in } \Omega\\
u_p=0 &\text{on } \partial\Omega,
\end{cases}
$$
where $\Omega$ is a bounded open set of ${\mathbb R}^N$ $(N\ge 2)$ with
Lipschitz boundary and $p > 1$. As far as the datum $f$ is
concerned, we analyze several cases: the most general one is $f\in
W^{-1,\infty}(\Omega)$. We also illustrate our results by means of
remarks and examples.