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

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 $\Delta_g=-\text{div}_g(\nabla_g)$. First, we prove that if either $\sup_M f$ and $\int_M a dv_g$ or $\sup_M a$ is sufficient small, the equation admits one positive smooth solution. Second, we show that the equation always admits one and only one positive smooth solution provided $\sup_M f \leqslant 0$. We should emphasize that we allow $a$ vanish somewhere. Along with these two results, existence and non-existence for related equations are also considered.