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 Δ1u:=div(Du|Du|) and having a singular term of the type f(x)uγ. Here f∈LN(Ω) is nonnegative, 00 almost everywhere, the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit one–dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of L∞–divergence–measure vector fields must be extended to deal with this equation.