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

We consider the singular elliptic problem of the form -Δ u + V(x)u = B(x)|u|2^*-2u + A(x)|u|2^*u, u∈ H¹(M), - Δ u + V ( x ) u = B ( x ) | u | 2 ∗ - 2 u + A ( x ) | u | 2 ∗ u , u ∈ H 1 ( M ) , where the coefficients are allowed to have low regularity. Under natural spectral assumptions on -Δ +V - Δ + V , geometric assumptions on the manifold M ensuring the Sobolev embedding H¹(M) L2^*(M) H 1 ( M ) ↪ L 2 ∗ ( M ) , and a suitable global integrability/smallness condition involving A A , B B , and a function ψ ∈ H¹(M) ψ ∈ H 1 ( M ) , we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and B≥ 0 B ≥ 0 , we obtain a positive finite-energy solution. The proof relies on a family of ε ε -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 A