The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$\{\begin{array}{ll}{\displaystyle -{\mathrm{\Delta }}_{1}u=h(u)f(x)}& \text{in }\mathrm{\Omega },\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(\mathrm{\Omega })$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.