Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the positive case (Q2928537)

Let \((M,g)\) be a smooth compact Riemannian manifold without boundary of dimension \(n\geq 3,\) and let \(f, h>0\) and \(a\geq 0\) be smooth functions on \(M\) with \(\int_M a d\text{vol}_g> 0.\) The authors prove two results involving the following partial differential equation which arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity NEWLINE\[NEWLINE \Delta_gu+hu=fu^{2^*-1}+\dfrac{a}{u^{2^*+1}}, NEWLINE\]NEWLINE where \(\Delta_g\) is the Laplace--Beltrami operator on \((M,g)\) and \(2^*=2n/(n-2).\)NEWLINENEWLINEThe first result shows that if \(\int_M a d\text{vol}_g\) is small enough, then the equation admits one positive smooth solution, while in the second one the condition on \(\int_M a d\text{vol}_g\) is relaxed to \(\sup_M f.\) As a by-product, the authors obtain a complete characterization of the existence of solutions in the case when \(\sup_M f\leq0.\)