The Dirichlet problem for the 1-Laplacian with a general singular term and L¹-data

We study the Dirichlet problem for an elliptic equation involving the 1-Laplace operator and a reaction term, namely: $\begin{cases}-{{\Delta}}_{1}u=h\left(u\right)f\left(x\right)\hfill & \text{in}\;{\Omega},\hfill \\ u=0\hfill & \text{on}\;\partial {\Omega},\hfill \end{cases}$ where ${\Omega}\subset {\mathbb{R}}^{N}$ is an open bounded set having Lipschitz boundary, f ∈ L¹(Ω) 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.