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 ${\mathrm{\Delta }}_{1}u$ and having a singular term of the type $\frac{f(x)}{u^{\gamma }}$. Here $f\in L^N(\mathrm{\Omega })$ is nonnegative, $0<\gamma \le 1$ and $\mathrm{\Omega }$ is a bounded domain with Lipschitz–continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions of $p$–Laplacian type problems. Moreover, when $f(x)>0$ 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 $L^{\mathrm{\infty }}$–divergence–measure vector fields must be extended to deal with this equation.