Prescribing a fourth order conformal invariant on the standard sphere, part II : blow up analysis and applications

Djadli, Zindine; Malchiodi, Andrea; Ould Ahmedou, Mohameden

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 2, pp. 387-434

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere $({𝕊}^n,h)$. We derive from this analysis some a priori estimates in dimension $5$ and $6$. On ${𝕊}^5$ these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On ${𝕊}^6$ we prove the existence of at least one solution when an index formula associated to this conformal invariant is different from zero.

Classification: 53C21, 35B45, 35J60, 53A30, 58G30

[1] S. I. Andersson - H. D. Doebner, Nonlinear partial differential operators and quantization procedures, Proceedings of a workshop held at the Technische Universitt Clausthal, S. I. Andersson - H.-D. Doebner Clausthal (eds.), July 1981. | Zbl

[2] A. Ambrosetti - J. Garcia Azorero - A. Peral, Perturbation of $\Delta u+u^{\frac{(N+2)}{(N-2)}}=0$, the Scalar Curvature Problem in ${ℝ}^N$ and related topics, Journal of Functional Analysis 165 (1999), 117-149. | Zbl | MR

[3] A. Arapostathis - M. F. Ghosh - S. Marcus, Harnack's inequality for cooperative weakly coupled systems, Comm. Partial Differential Equations 24 (1999), 1555-1571. | Zbl | MR

[4] T. Aubin, Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, Journal of Functional Analysis 32 (1979), 148-174. | Zbl | MR

[5] T. Aubin, “Some Nonlinear Problems in Differential Geometry”, Springer-Verlag, 1998. | MR

[6] T. Aubin - A. Bahri, Méthodes de topologie algébrique pour le probléme de la courbure scalaire prescrite, Journal des Mathématiques Pures et Appliquées 76 (1997), 525-549. | Zbl | MR

[7] S. Axler - P. Bourdon - W. Ramey, “Harmonic Function Theory”, Springer-Verlag, GTM 137, 1992. | Zbl | MR

[8] A. Bahri, “Critical points at infinity in some variational problems”, Research Notes in Mathematics, 182, Longman-Pitman, London, 1989. | Zbl | MR

[9] A. Bahri, An invariant for Yamabe type flows with applications to scalar curvature problems in higher dimensions, Duke Mathematical Journal 81 (1996), 323-466. | Zbl | MR

[10] A. Bahri - J. M. Coron, The Scalar-Curvature problem on the standard three-dimensional sphere, Journal of Functional Analysis 95 (1991), 106-172. | Zbl | MR

[11] M. Ben Ayed - Y. Chen - H. Chtioui - M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Mathematical Journal 84 (1996), 633-677. | Zbl | MR

[12] T. P. Branson, Group representations arising from Lorentz conformal geometry, Journal of Functional Analysis 74 (1987), 199-291. | Zbl | MR

[13] T. P. Branson, Differential operators canonically associated to a conformal structure, Mathematica Scandinavica 57-2 (1985), 293-345. | Zbl | MR

[14] T. P. Branson - S. Y. A. Chang - P. C. Yang, Estimates and extremal problems for the log-determinant on $4$-manifolds, Communications in Mathematical Physics 149 (1992), 241-262. | Zbl | MR

[15] S. Y. A. Chang, On a fourth order PDE in conformal geometry, Preprint, 1997.

[16] S. Y. A. Chang - M. J. Gursky - P. C. Yang, The scalar curvature equation on 2- and 3- spheres, Calculus of Variations and Partial Differential Equations 1 (1993), 205-229. | Zbl | MR

[17] S. Y. A. Chang - M. J. Gursky - P. C. Yang, Regularity of a fourth order non-linear PDE with critical exponent, American Journal of Mathematics 121 (1999), 215-257. | Zbl | MR

[18] S. Y. A. Chang - J. Qing - P. C. Yang, On the Chern-Gauss-Bonnet integral for conformal metrics on ${ℝ}^4$, Duke Mathematical Journal, to appear. | Zbl | MR

[19] S. Y. A. Chang - J. Qing - P. C. Yang, Compactification for a class of conformally flat $4$-manifolds, Inventiones Mathematicae, to appear. | Zbl | MR

[20] S. Y. A. Chang - P. Yang, A perturbation result in prescribing scalar curvature on ${𝕊}^n$, Duke Mathematical Journal 64 (1991), 27-69. | Zbl | MR

[21] S. Y. A. Chang - P. C. Yang, Extremal metrics of zeta functional determinants on $4$-manifolds, Annals of Mathematics 142 (1995), 171-212. | Zbl | MR

[22] S. Y. A. Chang - P. C. Yang, On a fourth order curvature invariant, In: “Spectral Problems in Geometry and Arithmetic”, T. Branson (ed.), Comtemporary Mathematics 237, AMS, 1999, 9-28. | Zbl | MR

[23] A. Connes, “Noncommutative geometry”, Academic Press, Inc., San Diego, CA, 1994. | Zbl | MR

[24] Z. Djadli - E. Hebey - M. Ledoux, Paneitz-type operators and applications, Duke Mathematical Journal 104 (2000) 129-169. | Zbl | MR

[25] Z. Djadli - A. Malchiodi - M. Ould Ahmedou, Prescribed fourth order conformal invariant on the standard sphere, Part I, to appear in Comm. Contem. Math. | Zbl

[26] H. D. Doebner - J. D. Hennig, Differential geometric methods in mathematical physics, Proceedings of the twelfth international conference held at the Technical University of Clausthal, H.-D. Doebner - J. D. Hennig (ed.). Clausthal, August 30-September 2, 1983. | Zbl

[27] J. Escobar - R. Schoen, Conformal metrics with prescribed scalar curvature, Inventiones Mathematicae 86 (1986), 243-254. | Zbl | MR

[28] D. Gilbarg - N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd edition, Springer-Verlag, 1983. | Zbl | MR

[29] M. J. Gursky, The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics 148 (1998), 315-337. | Zbl | MR

[30] E. Hebey, Changements de métriques conformes sur la sphère - Le problème de Nirenberg, Bull. Sci. Math. 114 (1990), 215-242. | Zbl | MR

[31] Y. Y. Li, Prescribing scalar curvature on ${𝕊}^n$ and related topics, Part I, Journal of Differential Equations 120 (1995), 319-410; Part II, Existence and compactness, Communications in Pure and Applied Mathematics, 49 (1996), 437-477. | Zbl | MR

[32] C. S. Lin, A classification of solutions of conformally invariant fourth order equation in ${ℝ}^n$, Commentari Mathematici Helveticii 73 (1998), 206-231. | Zbl | MR

[33] J. Moser, On a nonlinear problem in differential geometry, In: “Dynamical Systems”, M. Peixoto (ed.), Academic Press, New York, 1973, 273-280. | Zbl | MR

[34] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.

[35] S. Paneitz, Essential unitarization of symplectics and applications to field quantization, Journal of Functional Analysis 48 (1982), 310-359. | Zbl | MR

[36] M. H. Protter - H. F. Weinberger, “Maximum Principles in Differential Equations”, Springer-Verlag, 2nd edition, 1984. | Zbl | MR

[37] R. Schoen, On the number of constant scalar curvature metrics in a conformal class, In: “Differential Geometry: A Symposium in Honor of Manfredo Do Carmo”, H. B. Lawson - K. Tenenblat (eds.), (1991), 331-320, Wiley, New York. | Zbl | MR

[38] R. Schoen - D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calculus of Variations and Partial Differential Equations, 4 (1996), 1-25. | Zbl | MR

[39] J. Serrin - H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential and Integral Equations 9 (1996), 635-653. | Zbl | MR

[40] J. Wei - X. Xu, On conformal deformations of metrics on ${𝕊}^n$, Journal of Functional Analysis 157 (1998), 292-325. | Zbl | MR