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

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