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

The author was supported by a FNRS CdR grant J.0135.19, by the Fonds Thélam and by the ULB ARC grant "Partial Differential Equations in Interaction". The author warmly thanks the anonymous referee whose insightful comments greatly improved the readability of the paper

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  • 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 $.

    Mathematics Subject Classification: Primary: 35J60, 35B44, 35B40; Secondary: 35Q75.


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