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

TL;DR

This paper establishes the existence of finite-energy solutions to a class of singular elliptic equations on complete Riemannian manifolds, using regularization, variational methods, and geometric analysis.

Contribution

It introduces new existence results for Einstein-scalar field equations with low regularity coefficients on noncompact manifolds, including a nonexistence criterion.

Findings

Existence of nonnegative finite-energy supersolutions under spectral and geometric conditions.

Positive finite-energy solutions exist when Ricci curvature is nonnegative and coefficients are nonnegative.

A necessary integrability condition on the coefficient a7 for supersolution existence.

Abstract

We consider the singular elliptic problem of the form \[ -\Delta u + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] 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.