On the extreme value of Nehari manifold for nonlocal singular Schr{ö}dinger-Kirchhoff equations in ${\mathbb{R}}^N$

This article investigates the existence, non-existence, and multiplicity of weak solutions for a parameter-dependent nonlocal Schrödinger-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 ${\lambda }_{\ast }$.