The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Abstract

We establish some new existence and multiplicity results for positive solutions of the following Einstein-scalar field Lichnerowicz equations on compact manifolds $(M,g)$ without the boundary of dimension $n \geqslant 3$,

\[-\Delta_g u + hu = fu^\frac{n+2}{n-2} + au^{-\frac{3n-2}{n-2}},\]

with either a negative, a zero, or a positive Yamabe-scalar field conformal invariant $h$. These equations arise from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity.

The variational method can be naturally adopted to the analysis of the Hamiltonian constraint equations. However, it arises analytical difficulty, especially in the case when the prescribed scalar curvature-scalar field function $f$ may change sign. To our knowledge, such a problem in its most generic case remains open.

Finally, we establish some Liouville type results for a wider class of equations with constant coefficients including the Einstein-scalar field Lichnerowicz equation with constant coefficients.