Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials

Abstract

In this paper, we investigate the singular Dirichlet problem \begin{equation}\label{abs1} \left\{ \begin{array}{rclr} -Δ_1u &=&\dfracλ{|x|}\hbox{Sgn}\,(u)+\dfrac{f}{u^γ}&\quad\mbox{in}\; Ω,\\[1ex] u &=&0& \quad\mbox{on}\; \partialΩ, \end{array} \right. \end{equation} where $Δ_1u=\hbox{div}\left(\frac{Du}{|Du|}\right)$ denotes the $1$-Laplacian operator, the parameters are $λ\in {R}$ and $γ>0$, and $f$ is a non-negative function belonging to the Lorentz space $L^{N,\infty}(Ω)$. Our main goal is to establish the existence of non-trivial solutions under the sharp restriction $λ 0$. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as $λ\geq N-1$ and $f$ is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as $p\to 1^+$, of the solutions to the approximating $p$-Laplace problems \begin{equation} \left\{ \begin{array}{rclc} -Δ_pu&=&\dfracλ{|x|^p}|u|^{p-2}u+\dfrac{f}{u^γ}&\quad\mbox{ in }\; Ω,\\[1ex] u&=&0&\quad\mbox{ on }\; \partialΩ. \end{array} \right. \end{equation} Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.