In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{ array}{ rclr} -\Delta_p u_p&=&\frac{ \lambda}{ |x|^p}|u_p|^{ p-2}u_p+f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on }\partial\Omega, \end{ array}\right. $$ where $\Omega$ is a bounded open subset of $\mathbb{ R}^N$ with Lipschitz boundary, $\lambda\in\mathbb{ R}^+$, and $f$ is a nonnegative datum in $L^{ N,\infty}(\Omega)$. Under sharp smallness assumptions on the data $\lambda$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\left\{ \begin{ array}{ rclr}- \Delta_{ 1} u &=& \frac{ \lambda}{ |x|}{ \rm Sgn}(u)+f & \text{ in}\, \Omega,\\ u&=&0 & \text{ on}\ \partial \Omega,\end{ array}\right. $$ where $\Delta_{ 1} u ={ \rm div}\left(\frac{ D u}{ |Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.
Categories: math.AP