Prescribing Q-curvature on higher dimensional spheres
Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 2, pp. 259-295.

We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.

DOI: 10.5802/ambp.207
Classification: 35J60,  53C21,  58J05
Keywords: Variational problems, lack of compactness, Q curvature, critical points at infinity
Khalil El Mehdi 1

1 Université de Nouakchott Faculté des Sciences et Techniques BP 5026, Nouakchott MAURITANIA
@article{AMBP_2005__12_2_259_0,
     author = {Khalil El Mehdi},
     title = {Prescribing $Q$-curvature on higher dimensional spheres},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {259--295},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {12},
     number = {2},
     year = {2005},
     doi = {10.5802/ambp.207},
     zbl = {05016092},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.207/}
}
TY  - JOUR
AU  - Khalil El Mehdi
TI  - Prescribing $Q$-curvature on higher dimensional spheres
JO  - Annales mathématiques Blaise Pascal
PY  - 2005
DA  - 2005///
SP  - 259
EP  - 295
VL  - 12
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.207/
UR  - https://zbmath.org/?q=an%3A05016092
UR  - https://doi.org/10.5802/ambp.207
DO  - 10.5802/ambp.207
LA  - en
ID  - AMBP_2005__12_2_259_0
ER  - 
%0 Journal Article
%A Khalil El Mehdi
%T Prescribing $Q$-curvature on higher dimensional spheres
%J Annales mathématiques Blaise Pascal
%D 2005
%P 259-295
%V 12
%N 2
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.207
%R 10.5802/ambp.207
%G en
%F AMBP_2005__12_2_259_0
Khalil El Mehdi. Prescribing $Q$-curvature on higher dimensional spheres. Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 2, pp. 259-295. doi : 10.5802/ambp.207. https://ambp.centre-mersenne.org/articles/10.5802/ambp.207/

[1] T. Aubin; A. Bahri Une hypothèse de topologie algebrique pour le problème de la courbure scalaire prescrite, J Math. Pures Appl., Volume 76 (1997), pp. 843-850 | DOI | MR | Zbl

[2] T. Aubin Some nonlinear problems in differential geometry, Springer-Verlag, New York, 1997

[3] A. Bahri; J.M. Coron On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain, Comm. Pure Appl. Math., Volume 41 (1988), pp. 255-294 | DOI | MR | Zbl

[4] A. Bahri; P. Rabinowitz Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non linéaire, Volume 8 (1991), pp. 561-649 | Numdam | MR | Zbl

[5] A. Bahri Critical point at infinity in some variational problems, Pitman Res. Notes Math Ser 182, Longman Sci. Tech. Harlow, 1989 | MR | Zbl

[6] A. Bahri An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimension, A celebration of J. F. Nash Jr., Duke Math. Journal, Volume 81 (1996), pp. 323-466 | DOI | MR | Zbl

[7] M. Ben Ayed; Y. Chen; H. Chtioui; M. Hammami On the prescribed scalar curvature problem on 4- manifolds, Duke Math. J., Volume 84 (1996), pp. 633-677 | DOI | MR | Zbl

[8] M. Ben Ayed; K. El Mehdi The Paneitz curvature problem on lower dimensional spheres, Preprint The Abdus Salam ICTP, Trieste, Italy, Volume 48 (2003) | MR | Zbl

[9] M. Ben Ayed; K. El Mehdi Existence of conformal metrics on spheres with prescribed Paneitz curvature, Manuscripta Math, Volume 114 (2004), pp. 211-228 | MR | Zbl

[10] M. Ben Ayed; M. Hammami Critical points at infinity in a fourth order elliptic problem with limiting exponent, Nonlinear Anal. TMA, Volume 59 (2004), pp. 891-916 | MR | Zbl

[11] T. P. Branson; S. A. Chang; P. C. Yang Estimates and extremal problems for the log-determinant on 4-manifolds, Comm. Math. Phys., Volume 149 (1992), pp. 241-262 | DOI | MR | Zbl

[12] T. P. Branson Differential operators canonically associated to a conformal structure, Math. Scand., Volume 57 (1985), pp. 293-345 | EuDML | MR | Zbl

[13] T. P. Branson Group representations arising from Lorentz conformal geometry, J. Funct. Anal., Volume 74 (1987), pp. 199-291 | DOI | MR | Zbl

[14] H. Brezis; J.M. Coron Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., Volume 89 (1985), pp. 21-56 | DOI | MR | Zbl

[15] S. A. Chang; M. J. Gursky; P. C. Yang Regularity of a fourth order nonlinear PDE with critical exponent, Amer. J. Math., Volume 121 (1999), pp. 215-257 | DOI | MR | Zbl

[16] S. A. Chang; J. Qing; P. C. Yang Compactification for a class of conformally flat 4-manifolds, Invent. Math., Volume 142 (2000), pp. 65-93 | DOI | MR | Zbl

[17] S. A. Chang; J. Qing; P. C. Yang On the Chern-Gauss-Bonnet integral for conformal metrics on R4, Duke Math. J., Volume 103 (2000), pp. 523-544 | DOI | MR | Zbl

[18] S. A. Chang; P. C. Yang On a fourth order curvature invariant, Spectral problems in Geometry and Arithmetic, Contemporary Math., Volume 237 (1999), pp. 9-28 | Zbl

[19] S. A. Chang On Paneitz operator - fourth order differential operator in conformal geometry, Harmonic Analysis and PDE; Essays in Honor of A. P. Calderon, Editors: M. Christ, C. Kenig and C. Sadorsky; Chicago Lectures in Mathematics, Volume Chapter 8 (1999), pp. 127-150 | Zbl

[20] H. Chtioui Prescribing the scalar curvature problem on three and four manifolds, Adv. Nonlinear Stud., Volume 3 (2003), pp. 457-469 | MR | Zbl

[21] Z. Djadli; E. Hebey; M. Ledoux Paneitz-type operators and applications, Duke Math. J., Volume 104 (2000), pp. 129-169 | DOI | MR | Zbl

[22] Z. Djadli; A. Malchiodi; M. Ould Ahmedou Prescribing a fourth order conformal invariant on the standard sphere, Part I: a perturbation result, Commun. Contemp. Math., Volume 4 (2002), pp. 357-408 | DOI | MR | Zbl

[23] Z. Djadli; A. Malchiodi; M. Ould Ahmedou Prescribing a fourth order conformal invariant on the standard sphere, Part II: blow up analysis and applications, Annali della Scuola Normale Sup. di Pisa, Volume 5 (2002), pp. 387-434 | EuDML | Numdam | MR | Zbl

[24] P. Esposito; F. Robert Mountain pass critical points for Paneitz-Branson operators, Calc. Var. Partial Differential Equations, Volume 15 (2002), pp. 493-517 | DOI | MR | Zbl

[25] V. Felli Existence of conformal metrics on Sn with prescribed fourth-order invariant, Adv. Differential Equations, Volume 7 (2002), pp. 47-76 | MR | Zbl

[26] M. J. Gursky The Weyl functional, de Rham cohomology and Khaler-Einstein metrics, Ann. of Math., Volume 148 (1998), pp. 315-337 | DOI | MR | Zbl

[27] C. S. Lin A classification of solutions of a conformally invariant fourth order equation in Rn, Commentari Mathematici Helvetici, Volume 73 (1998), pp. 206-231 | DOI | MR | Zbl

[28] P. L. Lions The concentration compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, Volume 1 (1985), p. I: 165-201; II: 45-121 | EuDML | MR | Zbl

[29] J. Milnor Lectures on h-Cobordism Theorem, Princeton University Press, Princeton, 1965 | MR | Zbl

[30] S. Paneitz A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (1983) (Preprint)

[31] M. Struwe A global compactness result for elliptic boundary value problems involving nonlinearities, Math. Z., Volume 187 (1984), pp. 511-517 | DOI | EuDML | MR | Zbl

[32] J. Wei; X. Xu On conformal deformations of metrics on Sn, J. Funct. Anal., Volume 157 (1998), pp. 292-325 | DOI | MR | Zbl

[33] X. Xu; P. C. Yang Positivity of Paneitz operators, Discrete Contin. Dyn. Syst., Volume 7 (2001), pp. 329-342 | DOI | MR | Zbl

Cited by Sources: