Existence and classification of solutions for singular 1-Laplacian equations with gradient terms

We study the existence and qualitative properties of non-negative solutions to singular 1-Laplacian problems of the form \begin{equation*} \begin{cases} -Δ_1 u = g(u)|D u| + h(u)f & $\text{in }$ Ω,\newline u=0 & $\text{on }$ \partialΩ, \end{cases} \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, $0<f\in L^{N,\mathrm{\infty }}(\mathrm{\Omega })$, and $g,h$ are continuous positive functions that may blow up at the origin. We establish the existence of a positive bounded solution even when $g$ is non-integrable at zero and $h(0)=+\mathrm{\infty }$, breaking the non-existence phenomenon appearing in the associated $p$-Laplacian problems. Furthermore, we prove that solutions possess a non-zero boundary trace on a set of positive measure of $\mathrm{\partial }\mathrm{\Omega }$ whenever $g(s)$ grows sufficiently fast near $s=0$; by contrast, when this growth condition fails, solutions vanishing on the boundary may appear. Finally, when $h(0)<+\mathrm{\infty }$, we provide a sharp threshold classification of trivial and non-trivial solutions depending on the size of $f$.