Multiplicity of solutions for singular elliptic problems with Stein-Weiss term

In the present work, we establish the existence and multiplicity of positive solutions for the singular elliptic equations with a double weighted nonlocal interaction term defined in the whole space ${\mathbb{R}}^N$. The nonlocal term and the fact that the energy functional is not differentiable are the main difficulties for this kind of problem. We apply the Nehari method and the nonlinear Rayleigh quotient to prove that our main problem has at least two positive weak solutions. Furthermore, we prove a nonexistence result related to the extreme ${\lambda }^{\ast }>0$ given by the nonlinear Rayleigh quotient.