An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoff PDE with strong singularities. A byproduct is that −2 is no longer the critical position for the existence of positive solutions of PDE's with singular potentials and negative powers of the form: −|x| α ∆u = u −γ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain of R N containing 0, with N ≥ 3, α ∈ (0, N ) and −γ ∈ (−3, −1).