In this paper we analyze the asymptotic behaviour as \(p\rightarrow 1^+\) of solutions \(u_p\) to \( \left\{ \begin{array}{rclr} -\Delta _pu_p& =& \lambda |\nabla u_p|^{p-2}\nabla u_p\cdot \frac{x}{|x|^2}+ f& \quad \text{ in } \Omega ,\\ u_p& =& 0 & \quad \text{ on } \partial \Omega , \end{array}\right. \) where \(\Omega \) is a bounded open subset of \(\mathbb {R}^N\) with Lipschitz boundary containing the origin, \(\lambda \in \mathbb {R}\) , and f is a nonnegative datum in \(L^{N,\infty }(\Omega )\) . As a consequence, under suitable smallness assumptions on f and \(\lambda \) , we show sharp existence results of bounded solutions to the Dirichlet problems \({\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = \lambda \frac{D u}{|D u|}\cdot \frac{x}{|x|^2}+f & \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \) where \(\displaystyle \Delta _{1}u=\hbox {div}\,\left( \frac{Du}{|Du|}\right) \) is the 1-Laplacian operator. The case of a generic drift term in \(L^{N,\infty }(\Omega )\) is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.
