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Existence of conformal metrics with prescribed Q-curvature. (English) Zbl 1275.53035

Summary: We consider the problem of existence of conformal metrics with prescribed Q-curvature on the standard sphere \(S^n\), \(n \geq 5\). Under the assumption that the order of flatness at critical points of the prescribed Q-curvature function \(K(x)\) is \(\beta \in ]1, n - 4]\), we give precise estimates on the losses of compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.

MSC:

53C20 Global Riemannian geometry, including pinching
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[1] Paneitz, S. M., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 4, article 036 (2008) · Zbl 1145.53053 · doi:10.3842/SIGMA.2008.036
[2] Branson, T. P., Differential operators canonically associated to a conformal structure, Mathematica Scandinavica, 57, 2, 293-345 (1985) · Zbl 0596.53009
[3] Djadli, Z.; Hebey, E.; Ledoux, M., Paneitz-type operators and applications, Duke Mathematical Journal, 104, 1, 129-169 (2000) · Zbl 0998.58009 · doi:10.1215/S0012-7094-00-10416-4
[4] Chang, Sun-Y. A.; Yang, P. C., Extremal metrics of zeta function determinants on \(S^n\)-manifolds, Annals of Mathematics, 142, 1, 171-212 (1995) · Zbl 0842.58011 · doi:10.2307/2118613
[5] Djadli, Z.; Malchiodi, A., Existence of conformal metrics with constant \(Q\)-curvature, Annals of Mathematics, 168, 3, 813-858 (2008) · Zbl 1186.53050 · doi:10.4007/annals.2008.168.813
[6] Gursky, M. J., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Communications in Mathematical Physics, 207, 1, 131-143 (1999) · Zbl 0988.58013 · doi:10.1007/s002200050721
[7] Wei, J.; Xu, X., On conformal deformations of metrics on \(S^n\), Journal of Functional Analysis, 157, 1, 292-325 (1998) · Zbl 0924.58120 · doi:10.1006/jfan.1998.3271
[8] Abdelhedi, W.; Chtioui, H., On the prescribed Paneitz curvature problem on the standard spheres, Advanced Nonlinear Studies, 6, 4, 511-528 (2006) · Zbl 1184.53043
[9] Bensouf, A.; Chtioui, H., Conformal metrics with prescribed \(Q\)-curvature on \(S^n\), Calculus of Variations and Partial Differential Equations, 41, 3-4, 455-481 (2011) · Zbl 1218.53041 · doi:10.1007/s00526-010-0372-9
[10] Chtioui, H.; Rigane, A., On the prescribed \(Q\)-curvature problem on \(S^n\), Comptes Rendus Mathématique. Académie des Sciences. Paris, 348, 11-12, 635-638 (2010) · Zbl 1193.53101 · doi:10.1016/j.crma.2010.03.018
[11] Chtioui, H.; Rigane, A., On the prescribed Q-curvature problem on \(S^n\), Journal of Functional Analysis, 261, 10, 2999-3043 (2011) · Zbl 1233.53007 · doi:10.1016/j.jfa.2011.07.017
[12] Djadli, Z.; Malchiodi, A.; Ahmedou, M. O., Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result, Communications in Contemporary Mathematics, 4, 3, 375-408 (2002) · Zbl 1023.58020 · doi:10.1142/S0219199702000695
[13] Djadli, Z.; Malchiodi, A.; Ahmedou, M. O., Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications, Annali della Scuola Normale Superiore di Pisa, 1, 2, 387-434 (2002) · Zbl 1150.53012
[14] Felli, V., Existence of conformal metrics on \(S^n\) with prescribed fourth-order invariant, Advances in Differential Equations, 7, 1, 47-76 (2002) · Zbl 1054.53061
[15] Bahri, A., Critical Points at Infinity in Some Variational Problems. Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182, vi+I15+307 (1989), Harlow, UK: Longman Scientific & Technical, Harlow, UK · Zbl 0676.58021
[16] Bahri, A., An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension, Duke Mathematical Journal, 81, 2, 323-466 (1996) · Zbl 0856.53028 · doi:10.1215/S0012-7094-96-08116-8
[17] Lin, C.-S., A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\), Commentarii Mathematici Helvetici, 73, 2, 206-231 (1998) · Zbl 0933.35057 · doi:10.1007/s000140050052
[18] Ben Ayed, M.; Chen, Y.; Chtioui, H.; Hammami, M., On the prescribed scalar curvature problem on \(S^n\)-manifolds, Duke Mathematical Journal, 84, 3, 633-677 (1996) · Zbl 0862.53034 · doi:10.1215/S0012-7094-96-08420-3
[19] Hebey, E.; Robert, F., Asymptotic analysis for fourth order Paneitz equations with critical growth, Advances in Calculus of Variations, 4, 3, 229-275 (2011) · Zbl 1251.58005 · doi:10.1515/ACV.2011.001
[20] Ben Ayed, M.; El Mehdi, K., The Paneitz curvature problem on lower-dimensional spheres, Annals of Global Analysis and Geometry, 31, 1, 1-36 (2007) · Zbl 1170.35394 · doi:10.1007/s10455-005-9003-7
[21] Ben Mahmoud, R.; Chtioui, H., Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete and Continuous Dynamical Systems A, 32, 5, 1857-1879 (2012) · Zbl 1242.58006 · doi:10.3934/dcds.2012.32.1857
[22] Bahri, A.; Rabinowitz, P. H., Periodic solutions of Hamiltonian systems of three body type, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 8, 6, 561-649 (1991) · Zbl 0745.34034
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