Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to 1^{+}$

In this paper we analyze the asymptotic behaviour as $p\to 1^{+}$ of solutions $u_p$ to $$\{\begin{array}{rclr}-{\mathrm{\Delta }}_p{u}_p& =& \frac{\lambda }{\mathrm{\mid }x{\mathrm{\mid }}^p}\mathrm{\mid }{u}_p{\mathrm{\mid }}^{p-2}{u}_p+f& in\mathrm{\Omega },\\ u_p& =& 0& on\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^N$ with Lipschitz boundary, $\lambda \in {\mathbb{R}}^{+}$, and $f$ is a nonnegative datum in $L^{N,\mathrm{\infty }}(\mathrm{\Omega })$. 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 $$\{\begin{array}{rclr}-{\mathrm{\Delta }}_{1}u& =& \frac{\lambda }{\mathrm{\mid }x\mathrm{\mid }}\mathrm{S}\mathrm{g}\mathrm{n}(u)+f& \text{in}\mathrm{\Omega },\\ u& =& 0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where ${\mathrm{\Delta }}_{1}u=\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.