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 (Formula Presenetd) where Ω ⊂ ℝN is a bounded open set with Lipschitz boundary, f ∈ Lm(Ω) 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 (Ω) under sharp conditions on the summability of f and on the behavior 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 behavior of h at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna ([On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991) 721-730]) to the case of the 1-Laplacian as principal operator.