@article {,
	title = {Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the null case},
	journal = {Communications in Mathematical Physics},
	volume = {334},
	year = {2015},
	pages = {193{\textendash}222},
	abstract = {<p>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 $\int_M f dv_g \&lt; 0$, $\sup_M f\&gt;0$, and $\int_M a dv_g\&gt;0$. 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<br />
\[ \Delta _g u = f u^{2^\star - 1} + \frac{a}{u^{2^\star +1}}, \]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.</p>
},
	issn = {0010-3616},
	doi = {10.1007/s00220-014-2133-7},
	url = {http://dx.doi.org/10.1007/s00220-014-2133-7}
}
