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

 Access to full text

 Full (PDF)

1. [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. Zbl0513.00009
2. [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. Zbl0938.35056MR1696454
3. [3] A. Arapostathis – M. F. Ghosh – S. Marcus, Harnack’s inequality for cooperative weakly coupled systems, Comm. Partial Differential Equations 24 (1999), 1555-1571. Zbl0934.35039MR1708101
4. [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. Zbl0411.46019MR534672
5. [5] T. Aubin, “Some Nonlinear Problems in Differential Geometry”, Springer-Verlag, 1998. MR1636569
6. [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. Zbl0886.58109MR1465609
7. [7] S. Axler – P. Bourdon – W. Ramey, “Harmonic Function Theory”, Springer-Verlag, GTM 137, 1992. Zbl0765.31001MR1184139
8. [8] A. Bahri, “Critical points at infinity in some variational problems”, Research Notes in Mathematics, 182, Longman-Pitman, London, 1989. Zbl0676.58021MR1019828
9. [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. Zbl0856.53028MR1395407
10. [10] A. Bahri – J. M. Coron, The Scalar-Curvature problem on the standard three-dimensional sphere, Journal of Functional Analysis 95 (1991), 106-172. Zbl0722.53032MR1087949
11. [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. Zbl0862.53034MR1408540
12. [12] T. P. Branson, Group representations arising from Lorentz conformal geometry, Journal of Functional Analysis 74 (1987), 199-291. Zbl0643.58036MR904819
13. [13] T. P. Branson, Differential operators canonically associated to a conformal structure, Mathematica Scandinavica 57-2 (1985), 293-345. Zbl0596.53009MR832360
14. [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. Zbl0761.58053MR1186028
15. [15] S. Y. A. Chang, On a fourth order PDE in conformal geometry, Preprint, 1997. 
16. [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. Zbl0822.35043MR1261723
17. [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. Zbl0921.35032MR1680337
18. [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. Zbl0971.53028MR1763657
19. [19] S. Y. A. Chang – J. Qing – P. C. Yang, Compactification for a class of conformally flat $4$-manifolds, Inventiones Mathematicae, to appear. Zbl0990.53026MR1784799
20. [20] S. Y. A. Chang – P. Yang, A perturbation result in prescribing scalar curvature on ${𝕊}^n$, Duke Mathematical Journal 64 (1991), 27-69. Zbl0739.53027MR1131392
21. [21] S. Y. A. Chang – P. C. Yang, Extremal metrics of zeta functional determinants on $4$-manifolds, Annals of Mathematics 142 (1995), 171-212. Zbl0842.58011MR1338677
22. [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. Zbl0982.53035MR1710786
23. [23] A. Connes, “Noncommutative geometry”, Academic Press, Inc., San Diego, CA, 1994. Zbl0818.46076MR1303779
24. [24] Z. Djadli – E. Hebey – M. Ledoux, Paneitz-type operators and applications, Duke Mathematical Journal 104 (2000) 129-169. Zbl0998.58009MR1769728
25. [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. Zbl1023.58020
26. [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. Zbl0564.00012
27. [27] J. Escobar – R. Schoen, Conformal metrics with prescribed scalar curvature, Inventiones Mathematicae 86 (1986), 243-254. Zbl0628.53041MR856845
28. [28] D. Gilbarg – N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd edition, Springer-Verlag, 1983. Zbl0562.35001MR737190
29. [29] M. J. Gursky, The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics 148 (1998), 315-337. Zbl0949.53025MR1652920
30. [30] E. Hebey, Changements de métriques conformes sur la sphère – Le problème de Nirenberg, Bull. Sci. Math. 114 (1990), 215-242. Zbl0713.53023MR1056162
31. [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. Zbl0827.53039MR1347349
32. [32] C. S. Lin, A classification of solutions of conformally invariant fourth order equation in ${ℝ}^n$, Commentari Mathematici Helveticii 73 (1998), 206-231. Zbl0933.35057MR1611691
33. [33] J. Moser, On a nonlinear problem in differential geometry, In: “Dynamical Systems”, M. Peixoto (ed.), Academic Press, New York, 1973, 273-280. Zbl0275.53027MR339258
34. [34] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983. 
35. [35] S. Paneitz, Essential unitarization of symplectics and applications to field quantization, Journal of Functional Analysis 48 (1982), 310-359. Zbl0499.47025MR678176
36. [36] M. H. Protter – H. F. Weinberger, “Maximum Principles in Differential Equations”, Springer-Verlag, 2nd edition, 1984. Zbl0549.35002MR762825
37. [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. Zbl0733.53021MR1173050
38. [38] R. Schoen – D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calculus of Variations and Partial Differential Equations, 4 (1996), 1-25. Zbl0843.53037MR1379191
39. [39] J. Serrin – H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential and Integral Equations 9 (1996), 635-653. Zbl0868.35032MR1401429
40. [40] J. Wei – X. Xu, On conformal deformations of metrics on ${𝕊}^n$, Journal of Functional Analysis 157 (1998), 292-325. Zbl0924.58120MR1637945

1. Hans-Christoph Grunau, Mohameden Ould Ahmedou, Wolfgang Reichel, The Paneitz equation in hyperbolic space
2. Khalil El Mehdi, Prescribing $Q$-curvature on higher dimensional spheres
3. Zindine Djadli, Opérateurs géométriques et géométrie conforme