Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$

Abstract: 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.