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November  2021, 41(11): 5087-5103. doi: 10.3934/dcds.2021069

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

Université Libre de Bruxelles, Service d'Analyse CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

Received  November 2020 Revised  February 2021 Published  November 2021 Early access  April 2021

Fund Project: 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

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 $
.
Citation: Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5087-5103. doi: 10.3934/dcds.2021069
References:
[1]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundlehren der Mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[2]

R. Bartnik and J. Isenberg, The constraint equations, The Einstein Equations and the Large Scale Behavior of Gravitational Fields, Birkhäuser, Basel, (2004), 1-38.

[3]

P. Bizoń, S. Pletka and W. Simon, Initial data for rotating cosmologies, Classical Quantum Gravity, 32 (2015), 175015, 21 pp. doi: 10.1088/0264-9381/32/17/175015.

[4]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[5]

Y. Choquet-Bruhat and R. Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., 14 (1969), 329-335.  doi: 10.1007/BF01645389.

[6]

P. T. Chruściel and R. Gicquaud, Bifurcating solutions of the Lichnerowicz equation, Ann. Henri Poincaré, 18 (2017), 643-679.  doi: 10.1007/s00023-016-0501-x.

[7]

O. Druet and E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z., 263 (2009), 33-67.  doi: 10.1007/s00209-008-0409-3.

[8]

O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraint system, Math. Ann., 362 (2015), 839-886.  doi: 10.1007/s00208-014-1145-0.

[9]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143. Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[10]

J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.

[11]

Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edition, Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

[12]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134.

[13]

E. HebeyF. Pacard and D. Pollack, A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Comm. Math. Phys., 278 (2008), 117-132.  doi: 10.1007/s00220-007-0377-1.

[14]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.

[15]

B. Premoselli, Effective multiplicity for the Einstein-scalar field Lichnerowicz equation, Calc. Var. Partial Differential Equations, 53 (2015), 29-64.  doi: 10.1007/s00526-014-0740-y.

[16]

B. Premoselli, Stability and instability of the Einstein-Lichnerowicz constraint system, Int. Math. Res. Not. IMRN, (2016), 1951-2025. doi: 10.1093/imrn/rnv193.

[17]

B. Premoselli and J. Wei, Non-compactness and infinite number of conformal initial data sets in high dimensions, J. Funct. Anal., 270 (2016), 718-747.  doi: 10.1016/j.jfa.2015.06.018.

[18]

F. Robert, Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d'ordre deux, http://www.iecn.u-nancy.fr/ frobert/ConstrucGreen.pdf.

[19]

R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76, http://projecteuclid.org/euclid.cmp/1103904790. doi: 10.1007/BF01940959.

show all references

References:
[1]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundlehren der Mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[2]

R. Bartnik and J. Isenberg, The constraint equations, The Einstein Equations and the Large Scale Behavior of Gravitational Fields, Birkhäuser, Basel, (2004), 1-38.

[3]

P. Bizoń, S. Pletka and W. Simon, Initial data for rotating cosmologies, Classical Quantum Gravity, 32 (2015), 175015, 21 pp. doi: 10.1088/0264-9381/32/17/175015.

[4]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[5]

Y. Choquet-Bruhat and R. Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., 14 (1969), 329-335.  doi: 10.1007/BF01645389.

[6]

P. T. Chruściel and R. Gicquaud, Bifurcating solutions of the Lichnerowicz equation, Ann. Henri Poincaré, 18 (2017), 643-679.  doi: 10.1007/s00023-016-0501-x.

[7]

O. Druet and E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z., 263 (2009), 33-67.  doi: 10.1007/s00209-008-0409-3.

[8]

O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraint system, Math. Ann., 362 (2015), 839-886.  doi: 10.1007/s00208-014-1145-0.

[9]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143. Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[10]

J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.

[11]

Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd edition, Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

[12]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134.

[13]

E. HebeyF. Pacard and D. Pollack, A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Comm. Math. Phys., 278 (2008), 117-132.  doi: 10.1007/s00220-007-0377-1.

[14]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.

[15]

B. Premoselli, Effective multiplicity for the Einstein-scalar field Lichnerowicz equation, Calc. Var. Partial Differential Equations, 53 (2015), 29-64.  doi: 10.1007/s00526-014-0740-y.

[16]

B. Premoselli, Stability and instability of the Einstein-Lichnerowicz constraint system, Int. Math. Res. Not. IMRN, (2016), 1951-2025. doi: 10.1093/imrn/rnv193.

[17]

B. Premoselli and J. Wei, Non-compactness and infinite number of conformal initial data sets in high dimensions, J. Funct. Anal., 270 (2016), 718-747.  doi: 10.1016/j.jfa.2015.06.018.

[18]

F. Robert, Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d'ordre deux, http://www.iecn.u-nancy.fr/ frobert/ConstrucGreen.pdf.

[19]

R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76, http://projecteuclid.org/euclid.cmp/1103904790. doi: 10.1007/BF01940959.

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