Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the positive case

This is the third and last in our series of papers concerning solution of the Einstein-scalar field Lichnerowicz equations on Riemannian manifolds. Let $(M,g)$ be a smooth compact Riemannian manifold without the boundary of dimension $n \geqslant 3$, $f$, $h>0$, and $a \geqslant 0$ are smooth functions on $M$ with $\int_M a dv_g>0$. In this article, we prove two major results involving the following partial differential equation arising from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity
\[{\Delta _g}u + hu = f{u^{{2^\star} - 1}} + \frac{a}{{{u^{{2^\star} +1}}}},\]where $\Delta_g  = -\text{div}_g(\nabla_g \cdot )$. In the first part of the paper, we prove that if $\int_M a dv_g$ is sufficient small, the equation admits one positive smooth solution. In the second part of the paper, we show that the condition for $\int_M a dv_g$ can be relaxed if $\sup_M f$ is small. As a by-product of this result, we are able to get a complete characterization of the existence of solutions in the case when $\sup_M f \leqslant 0$. In addition to the two main results above, we should emphasize that we allow $a$ to have zeros in $M$.