In this paper we deal with some results concerning semilinear elliptic singular problems with Dirichlet boundary conditions. The problem becomes singular where the solution u vanishes. The model of this kind of problems is ------- u ≥ 0 in Ω, ------- - div A(x) Du = F(x, u) in Ω, ------- u = 0 on ∂Ω, ------- where Ω is a bounded open set of RN , N ≥ 1, A is a coercive matrix with coefficients in L^∞ (Ω) and F : (x, s) ∈ Ω x [0, + ∞[ -> F(x, s) ∈ [0, + ∞] is a Carathéodory function which is singular at s = 0. ------- Our aim is to study the meaning of the assumptions made on the singular function F(x,s) in the papers [D. Giachetti, P.J. Martinez Aparicio, F. Murat, A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization, J. Math. Pures et Appl., 107, (2017), pp. 41-77], [D. Giachetti, P.J. Martinez Aparicio, F. Murat, Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0, Ann. Sc. Norm. Sup. Pisa, accepted, (2017)], [D. Giachetti, P.J. Martinez Aparicio, F. Murat, Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes, submitted, (2017)], to extend some uniqueness results of the solution given in the same papers, and to prove the L^∞ regularity of the solutions under some regularity assumption on the data. ------- This paper has been published in "Trends in differential equations and applications", ed. by F. Ortegon Gallego, M.V. Redondo Neble and J.R. Rodriguez Galvan. SEMA-SIMAI Springer Series 8, Springer International Publishing Switzerland, (2016), pp. 221-241.