Skip to main content
SpringerLink
Log in

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions \(n = 3, 4, 5\). As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Allen, P.T., Clausen, A., Isenberg, J.: Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Classic. Quant. Grav. 25(7), 075009 (2008)

  2. Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 252. Springer, New York (1982)

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

  4. Beig, R., Chruściel, P.T., Schoen, R.: Kids are non-generic. Ann. Henri Poincaré 6(1), 155–194 (2005)

    Article  MATH  Google Scholar 

  5. Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth. Commun. Pure Appl. Math. 42(3), 271–297 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Choquet-Bruhat, Y., Isenberg, J., Pollack, D.: Applications of theorems of jean leray to the einstein-scalar field equations. J. Fixed Point Theory Appl. 1(1), 31–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Choquet-Bruhat, Y., Isenberg, J., Pollack, D.: The constraint equations for the einstein-scalar field system on compact manifolds. Classic. Quant. Grav. 24(4), 809–828 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dahl, M., Gicquaud, R., Humbert, E.: A limit equation associated to the solvability of the vacuum einstein constraint equations by using the conformal method. Duke Math. J. 161(14), 2669–2697 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Druet, O.: Compactness for yamabe metrics in low dimensions. Int. Math. Res. Notices 23, 1143–1191 (2004)

    Article  MathSciNet  Google Scholar 

  10. Druet, O., Hebey, E.: Stability and instability for einstein-scalar field lichnerowicz equations on compact riemannian manifolds. Math. Z. 263(1), 33–67 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fourès-Bruhat, Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hebey, E., Pacard, F., Pollack, D.: A variational analysis of einstein-scalar field lichnerowicz equations on compact riemannian manifolds. Commun. Math. Phys. 278(1), 117–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hebey, E., Veronelli, G.: The lichnerowicz equation in the closed case of the Einstein–Maxwell theory. Trans. Am. Math. Soc. (2011, accepted for publication)

  14. Holst, M., Nagy, G., Tsogtgerel, G.: Rough solutions of the einstein constraints on closed manifolds without near-cmc conditions. Commun. Math. Phys. 288(2), 547–613 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Isenberg, J.: Constant mean curvature solutions of the einstein constraint equations on closed manifolds. Classic. Quant. Grav. 12(9), 2249–2274 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, Y., Zhu, M.: Yamabe type equations on three-dimensional riemannian manifolds. Commun. Contemp. Math. 1(1), 1–50 (1999)

    Article  MathSciNet  Google Scholar 

  17. Lichnerowicz, A.: L’intégration des équations de la gravitation relativiste et le problème des \(n\) corps. J. Math. Pures Appl. 9(23), 37–63 (1944)

    MathSciNet  Google Scholar 

  18. Ma, L., Wei, J.: Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds. J. Math. Pures Appl. (2012). doi:10.1016/j.matpur.2012.06.009

  19. Maxwell, D.: A class of solutions of the vacuum einstein constraint equations with freely specified mean curvature. Math. Res. Lett. 16(4), 627–645 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ngô, Q.A., Xu, X.: Existence results for the einstein-scalar field lichnerowicz equations on compact riemannian manifolds. Adv. Math. 230(4–6), 2378–2415 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Premoselli, B.: The Einstein-scalar field constraint system in the positive case. Commun. Math. Phys. 326(2), 543–557 (2014). doi:10.1007/s00220-013-1852-5

  22. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)

  23. Robert, F.: Existence et asymptotiques optimales des fonctions de green des opérateurs elliptiques d’ordre deux (2009, personal notes)

  24. Sattinger, D.H. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1971/1972)

  25. Schoen, R.M.: Lecture notes from courses at stanford (1988, preprint, written by D. Pollack)

  26. Schoen, R.M.: On the number of constant scalar curvature metrics in a conformal class. In: Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol 52, pp. 311–320. Longman Sci. Tech., Harlow (1991)

  27. Struwe, M.: Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 34. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edn. Springer, Berlin (2008)

Download references

Acknowledgments

The author warmly thanks Olivier Druet and Emmanuel Hebey for constant support and valuable remarks during the elaboration of this paper.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, AGM, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 , Cergy-Pontoise Cedex, France

    Bruno Premoselli

  2. Laboratoire de Mathématiques, UMPA, Ecole Normale Supérieure de Lyon (Site Sciences), 46 Allée d’Italie, 69364 , Lyon Cedex 07, France

    Bruno Premoselli

Authors
  1. Bruno Premoselli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Premoselli.

Additional information

Communicated by M. Struwe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-014-0740-y

Mathematics Subject Classification

Search

Navigation