Elliptic problems involving the 1-Laplacian and a singular lower order term

This paper is concerned with the Dirichlet problem for an equation involving the $1$--Laplacian operator $Delta_1 u:=Divleft(rac{Du}{|Du|} ight)$ and having a singular term of the type $rac{f(x)}{u^gamma}$. Here $fin L^N(Omega)$ is nonnegative, $00$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of $L^infty$--divergence--measure vector fields must be extended to deal with this equation.