Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3

On a closed $ 3 $-dimensional Riemannian manifold $ (M,g) $ we investigate the limit of the Einstein-Lichnerowicz equation

$ \begin{equation} \triangle_g u + h u = f u^5 + \frac{\theta a}{u^7} \end{equation} $

as the momentum parameter $ \theta \to 0 $. Under a positive mass assumption on $ \triangle_g +h $, we prove that sequences of positive solutions to this equation converge in $ C^2(M) $, as $ \theta \to 0 $, either to zero or to a positive solution of the limiting equation $ \triangle_g u + h u = f u^5 $. We also prove that the minimizing solution of (1) constructed by the author in [15] converges uniformly to zero as $ \theta \to 0 $.