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

Abstract

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.