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

Abstract

This paper is concerned with the Dirichlet problem for an equation involving the 1–Laplacian operator Δ 1 u : = div ( D u | D u | ) and having a singular term of the type f ( x ) u γ . Here f ∈ L N ( Ω ) is nonnegative, 0 < γ ⩽ 1 and Ω is an open bounded set with Lipschitz-continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions to p-Laplacian type problems. Moreover, when f ( x ) > 0 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.