Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in the Null Case

Abstract

This is the second in our series of papers concerning positive solutions of the Einstein-scalar field Lichnerowicz equations. Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension $${n \geqslant 3}$$ , f and $${a \geqslant 0}$$ are two smooth functions on M with $${\int_M f dv_g < 0}$$ , sup M f > 0, and $${\int_M a dv_g > 0}$$ . In this article, we prove two results involving the following equation arising from the Hamiltonian constraint equation for the Einstein-scalar field equation in general relativity $$\Delta _g u = f u^{2^\star - 1} + \frac{a}{u^{2^\star +1}},$$ where $${\Delta_g = -{\rm div}_g(\nabla_g)}$$ and $${2^{\star} = 2n/(n - 2)}$$ . First, we prove that if either sup M f and $${\int}$$ M a dv g or sup M a is sufficiently small, the equation admits one positive smooth solution. Second, we show that the equation always admits one and only one positive smooth solution provided $${{\rm sup}_M f \leqslant 0}$$ . We should emphasize that we allow a to vanish somewhere. Along with these two results, existence and non-existence for related equations are also considered.