Optimal global BV regularity for 1-Laplace type BVP's with singular lower order terms

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem equation* cases - ₁ u= h (u) f & in, \\ u=0 & on, cases equation* where N is a bounded open set with Lipschitz boundary, f Lᵐ () with m 1 is a nonnegative function and h R^+ R^+ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on h at zero, we prove existence of global finite energy solutions in BV () under sharp conditions on the summability of f and on the behaviour of h at infinity. Roughly speaking, the faster h goes to zero at infinity, the less regularity is required on f. In contrast to the p-Laplacian case (p>1), we show that the behaviour of h at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (lm) to the case of the 1-Laplacian as principal operator.