Skip to main content
SpringerLink
Log in

Conformal metrics with prescribed Q-curvature on S n

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

Abstract

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n ≥ 5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler–Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.

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

Similar content being viewed by others

References

  1. Abdelhedi W., Chtioui H.: On the prescribed Paneitz curvature problem on the standard spheres. Adv. Nonlinear Stud. N4, 511–528 (2006)

    MathSciNet  Google Scholar 

  2. Bahri, A.: Critical Point at Infinity in Some Variational Problems. Pitman Research Notes in Mathematics Series, vol. 182. Longman Scientific and Technical, Harlow (1989)

  3. Bahri A.: An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions. A celebration of J. F. Nash Jr. Duke Math. J. 81, 323–466 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bahri A., Coron J.M.: The scalar curvature problem on the standard three dimensional spheres. J. Funct. Anal. 95, 106–172 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahri A., Rabinowitz P.: Periodic orbits of Hamiltonian systems of three body type. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 561–649 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Ben Ayed M., El Mehdi K.: The Paneitz curvature problem on lower dimensional spheres. Ann. Glob. Anal. Geom. 31(1), 1–36 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Ayed M., Chen Y., Chtioui H., Hammami M.: On the prescribed scalar curvature problem on 4-manifolds. Duke Math. J. 84, 633–677 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Branson T.P.: Differential operators canonically associated to a conformal structure. Math. Scand. 57, 293–345 (1985)

    MathSciNet  MATH  Google Scholar 

  9. Chang S.Y.A., Yang P.C.: Extremal metrics of zeta function determinants on 4-manifolds. Ann. Math. 142, 171–212 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conley, C.C.: Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics, vol. 38. AMS, Providence (1978)

  11. Djadli Z., Malchiodi A.: Existence of conformal metrics with constant Q-curvature. Ann. Math. (2) 168(N3), 813–858 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Djadli Z., Hebey E., Ledoux M.: Paneitz type operators and application. Duke Math. J. 104(N1), 129–169 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Djadli Z., Malchiodi A., Ould Ahmadou M.: Prescribing a fourth order conformal invariant on the standard sphere. I: A perturbation result. Commun. Contemp. Math. 4, 357–405 (2002)

    Article  MathSciNet  Google Scholar 

  14. Djadli, Z., Malchiodi, A., Ould Ahmadou, M.: Prescribing a fourth order conformal invariant on the standard sphere. II: Blow up analysis and applications, Ann. Sci. Norm. Super Pisa 387–434 (2002)

  15. El Mehdi, K.: Prescribing Q-curvature on higher dimensional sphere. Ann. Math. Blaise Pascal 12(n2), 259–295 (2005)

    Google Scholar 

  16. Felli V.: Existence of conformal metrics on S n with prescribed fourth-order invariant. Adv. Differ. Equ. 7, 47–76 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Gursky M.: The principal eigen value of a conormally invariant differential operator, with an application to semilinear elliptic PDE. Commun. Math. Phys. 207, 131–147 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  19. Li Y.Y.: Prescribing scalar curvature on S n and related topics, Part I. J. Differ. Equ. 120, 319–410 (1995)

    Article  MATH  Google Scholar 

  20. Lin C.S.: A classification of solutions of a conformally invariant fourth order equation in \({\mathbb{R}^n}\) . Comment. Math. Helv. 73, 206–231 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Milnor J.: Lectures on the H-Cobordism Theorem. Princeton University Press, Princeton (1965)

    MATH  Google Scholar 

  22. Paneitz S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. SIGMA 4(036), 1–3 (2008)

    MathSciNet  Google Scholar 

  23. Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Schoen R., Zhang D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differ. Equ. 4, 1–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Van der Vost R.C.A.M.: Best constant for the embedding of the space \({H^{2}(\Omega) \cap H_{0}^{1}\subset L^{p+1}(\Omega)}\) . Differ. Integr. Equ. 6, 259–276 (1993)

    Google Scholar 

  26. Van der Vost R.C.A.M.: Fourth order elliptic equations with critical growth. C. R. Acad. Sci. Paris 320, 295–299 (1995)

    Google Scholar 

  27. Wei J., Xu X.: On conformal deformations of metrics on S n. J. Funct. Anal. 157, 292–325 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route Soukra, Sfax, Tunisia

    Aymen Bensouf & Hichem Chtioui

Authors
  1. Aymen Bensouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hichem Chtioui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hichem Chtioui.

Additional information

Communicated by A. Malchiodi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bensouf, A., Chtioui, H. Conformal metrics with prescribed Q-curvature on S n . Calc. Var. 41, 455–481 (2011). https://doi.org/10.1007/s00526-010-0372-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-010-0372-9

Mathematics Subject Classification (2000)

Search

Navigation