  This article investigates the existence, non-existence, and multiplicity of
weak solutions for a parameter-dependent nonlocal Schr\"odinger-Kirchhoff type
problem on $\mathbb R^N$ involving singular non-linearity. By performing fine
analysis based on Nehari submanifolds and fibre maps, our goal is to show the
problem has at least two positive solutions even if $\lambda$ lies beyond the
extremal parameter $\lambda_\ast$.
