Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds

We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold