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

Abstract We consider the singular elliptic problem of the form $$ -\Delta u + V(x)u = \mathcal {B}(x)|u|^{2^*-2}u + \frac{\mathcal {A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), $$ - 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 $$-\Delta +V$$ - Δ + V , geometric assumptions on the manifold M ensuring the Sobolev embedding $$H^1(M)\hookrightarrow L^{2^*}(M)$$ H 1 ( M ) ↪ L 2 ∗ ( M ) , and a suitable global integrability/smallness condition involving $$\mathcal {A}$$ A , $$\mathcal {B}$$ B , and a function $$\psi \in H^1(M)$$ ψ ∈ H 1 ( M ) , we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $$\mathcal {B}\ge 0$$ B ≥ 0 , we obtain a positive finite-energy solution. The proof relies on a family of $$\varepsilon $$ 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 $$\mathcal {A}$$