A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

TL;DR

This paper proves the existence of positive solutions to a Lichnerowicz equation in Einstein-scalar field theory on non-CMC closed manifolds, addressing supercritical and singular terms using fixed-point methods.

Contribution

It introduces new existence results for solutions in non-constant mean curvature settings with supercritical terms, expanding understanding of Einstein-scalar field equations.

Findings

Existence of positive, bounded solutions under certain conditions

Conditions preventing the existence of solutions

Application of fixed-point argument with sub- and supersolutions

Abstract

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.