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

We consider the singular elliptic problem of the form $$-\Delta u + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M),$$ where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-\Delta+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $\psi \in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.