Kirchhoff type equations with strong singularities

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{|^\alpha }\Delta u = {u^{{\rm{ - }}\gamma }}$ in $Ω$ , $u = 0$ on $\partial \Omega $ , where $\Omega$ is a bounded domain of ${\mathbb{R}}^{N}$ containing 0, with $N \ge 3$ , $\alpha \in \left( {0, N} \right)$ and $ - \gamma \in \left( { - 3, - 1} \right)$ .