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

Abstract

On a closed 3-dimensional Riemannian manifold (M, g) we investigate the limit of the Einstein-Lichnerowicz equation(1)as the momentum parameter θ → 0. Under a positive mass assumption on g + h, we prove that sequences of positive solutions to this equation converge in C 2 (M ), as θ → 0, either to zero or to a positive solution of the limiting equation g u + hu = f u 5 . We also prove that the minimizing solution of (1) constructed by the author in [15] converges uniformly to zero as θ → 0.