Skip to main content
SpringerLink
Log in

The Nirenberg problem and its generalizations: a unified approach

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations \(P_\sigma (v)= Kv^{\frac{n+2\sigma }{n-2\sigma }}\) on the standard unit sphere \(\mathbb {S}^n\) for all \(\sigma \in (0,n/2)\), where \(P_\sigma \) is the intertwining operator of order \(2\sigma \). Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When \(\sigma =1\), it is the prescribing scalar curvature problem or the Nirenberg problem, and when \(\sigma =2\), it is the prescribing Q-curvature problem.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abdelhedi, W., Chtioui, H., Hajaiej, H.: A complete study of the lack of compactness and existence results of a Fractional Nirenberg Equation via a flatness hypothesis: Part I. arXiv:1409.5884

  2. Bahri, A., Coron, J.-M.: The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 255, 106–172 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baird, P., Fardoun, A., Regbaoui, R.: \(Q\)-curvature flow on \(4\)-manifolds. Calc. Var. Partial Differ. Equ. 27(1), 75–104 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baird, P., Fardoun, A., Regbaoui, R.: Prescribed \(Q\)-curvature on manifolds of even dimension. J. Geom. Phys. 59(2), 221–233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. 138(2), 213–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourguignon, J.P., Ezin, J.P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Am. Math. Soc. 301, 723–736 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Branson, T.P.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Am. Math. Soc. 347, 367–3742 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Brendle, S.: Prescribing a higher order conformal invariant on \({\mathbb{S}^n}\). Commun. Anal. Geom. 11, 837–858 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brendle, S.: Global existence and convergence for a higher order flow in conformal geometry. Ann. Math. (2) 158(1), 323–343 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caffarelli, L.A., Jin, T., Sire, Y., Xiong, J.: Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities. Arch. Ration. Mech. Anal. 213(1), 245–268 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial. Differ. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Case, J., Chang, S.-Y.: On fractional GJMS operators. Commun. Pure Appl. Math. 69(6), 1017–1061 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chang, S.-Y., González, M.: Fractional Laplacian in conformal geometry. Adv. Math. 226, 1410–1432 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chang, S.-Y., Gursky, M.J., Yang, P.: The scalar curvature equation on \(2\)- and \(3\)-spheres. Calc. Var. Partial Differ. Equ. 1, 205–229 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chang, S.-Y., Gursky, M.J., Yang, P.: An equation of Monge–Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. Math. (2) 155(3), 709–787 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chang, S.-Y., Xu, X., Yang, P.: A perturbation result for prescribing mean curvature. Math. Ann. 310, 473–496 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chang, S.-Y., Yang, P.: Prescribing Gaussian curvature on \(\mathbb{S}^2\). Acta Math. 159, 215–259 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chang, S.-Y., Yang, P.: Conformal deformation of metrics on \(\mathbb{S}^2\). J. Differ. Geom. 27, 259–296 (1988)

    Article  MathSciNet  Google Scholar 

  19. Chang, S.-Y., Yang, P.: Extremal metrics of zeta function determinants on \(4\)-manifolds. Ann. Math. (2) 142(1), 171–212 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Y.-H., Liu, C., Zheng, Y.: Existence results for the fractional Nirenberg problem. arXiv:1406.2770

  21. Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, W., Li, C., Ou, B.: \(Q\)-curvature flow on the standard sphere of even dimension. J. Funct. Anal. 261(4), 934–980 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, W., Zhu, J.: Indefinite fractional elliptic problem and Liouville theorems. J. Differ. Equ. 260(5), 4758–4785 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Djadli, Z., Hebey, E., Ledoux, M.: Paneitz-type operators and applications. Duke Math. J. 104(1), 129–169 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Djadli, Z., Malchiodi, A., Ahmedou, M.O.: Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result. Commun. Contemp. Math. 4, 375–408 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Djadli, Z., Malchiodi, A., Ahmedou, M.O.: Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 387–434 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Djadli, Z., Malchiodi, A., Ahmedou, M.O.: The prescribed boundary mean curvature problem on \(\mathbb{B}^4\). J. Differ. Equ. 206, 373–398 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Escobar, J.F.: Conformal metrics with prescribed mean curvature on the boundary. Calc. Var. Partial Differ. Equ. 4, 559–592 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Escobar, J.F., Garcia, G.: Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. J. Funct. Anal. 211, 71–152 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Escobar, J.F., Schoen, R.: Conformal metrics with prescribed scalar curvature. Invent. Math. 86, 243–254 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  32. Esposito, P., Robert, F.: Mountain pass critical points for Paneitz–Branson operators. Calc. Var. Partial Differ. Equ. 15(4), 493–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Fang, F.: Infinitely many non-radial sign-changing solutions for a Fractional Laplacian equation with critical nonlinearity. arXiv:1408.3187

  34. Fefferman, C., Graham, C.R.: Conformal invariants. The mathematical heritage of Élie Cartan (Lyon, : Astérisque). Numero Hors Serie 1985, 95–116 (1984)

    Google Scholar 

  35. Fefferman, C., Graham, C.R.: Juhl’s formulae for GJMS operators and \(Q\)-curvatures. J. Am. Math. Soc. 26, 1191–1207 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. González, M., Mazzeo, R., Sire, Y.: Singular solutions of fractional order conformal Laplacians. J. Geom. Anal. 22(3), 845–863 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. González, M., Qing, J.: Fractional conformal Laplacians and fractional Yamabe problems. Anal. PDE. 6(7), 1535–1576 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian I. Exist. J. Lond. Math. Soc. 46, 557–565 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gursky, M., Malchiodi, A.: A strong maximum principle for the Paneitz operator and a non-local flow for the \(Q\)-curvature. J. Eur. Math. Soc. (JEMS) 17(9), 2137–2173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Han, Z.-C.: Prescribing Gaussian curvature on \(\mathbb{S}^2\). Duke Math. J. 61, 679–703 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  43. Han, Z.-C.: The Yamabe problem on manifolds with boundary: existence and compactness results. Duke Math. J. 99, 489–542 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hang, F.B., Yang, P.C.: \(Q\)-curvature on a class of \(3\) manifolds. Commun. Pure Appl. Math. 69(4), 734–744 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Hang, F.B., Yang, P.C.: \(Q\) curvature on a class of manifolds of dimension at least \(5\). Commun. Pure Appl. Math. 69(8), 1452–1491 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Hang, F.B., Yang, P.C.: Sign of Green’s function of Panetiz operators and the \(Q\) curvature. Int. Math. Res. Not. IMRN (19) (2015), 9775–9791

  47. Hebey, E., Robert, F.: Asymptotic analysis for fourth order Paneitz equations with critical growth. Adv. Calc. Var. 4(3), 229–275 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hebey, E., Robert, F., Wen, Y.: Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry. Commun. Contemp. Math. 8(1), 9–65 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Ho, P.T.: Prescribed \(Q\)-curvature flow on \(\mathbb{S}^n\). J. Geom. Phys. 62(5), 1233–1261 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ho, P.T.: Results of prescribing \(Q\)-curvature on \(\mathbb{S}^n\). Arch. Math. (Basel) 100(1), 85–93 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Jin, T., Li, Y.Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. (JEMS) 16(6), 1111–1171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. Jin, T., Li,Y.Y., Xiong, J.: On a fractional Nirenberg problem, part II: existence of solutions. Int. Math. Res. Not. IMRN, (6) (2015), 1555–1589

  53. Jin, T., Xiong, J.: A fractional Yamabe flow and some applications. J. Reine Angew. Math. 696, 187–223 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Jin, T., Xiong, J.: Sharp constants in weighted trace inequalities on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 48(3–4), 555–585 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. Juhl, A.: On the recursive structure of Branson’s \(Q\)-curvature. Math. Res. Lett. 21(3), 495–507 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Juhl, A.: \(Q\)-curvatures. Geom. Funct. Anal. 23(4), 1278–1370 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Koutroufiotis, D.: Gaussian curvature and conformal mapping. J. Differ. Geom. 7, 479–488 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  58. Li, J., Li, Y., Liu, P.: The \(Q\)-curvature on a 4-dimensional Riemannian manifold \((M, g)\) with \(\int _{M}Q{\rm d} V_g=8\pi ^2\). Adv. Math. 231(3–4), 2194–2223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Li, Y.Y.: Prescribing scalar curvature on \({\mathbb{S}^n}\) and related problems. I. J. Differ. Equ. 120, 319–410 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  60. Li, Y.Y.: Prescribing scalar curvature on \({\mathbb{S}^n}\) and related problems. II. Existence and compactness. Comm. Pure Appl. Math. 49, 541–597 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  61. Li, Y.Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6, 153–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  62. Li, Y.Y.; Xiong, J.: Compactness of conformal metrics with constant \(Q\)-curvature. I. arXiv:1506.00739

  63. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (2) 118, 349–374 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  64. Malchiodi, A., Struwe, M.: \(Q\)-curvature flow on \(\mathbb{S}^4\). J. Differ. Geom. 73(1), 1–44 (2006)

    Article  MATH  Google Scholar 

  65. Moser, J.: On a Nonlinear Problem in Differential Geometry. Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971). Academic Press, New York (1973)

    Google Scholar 

  66. Ndiaye, C.B.: Constant \(Q\)-curvature metrics in arbitrary dimension. J. Funct. Anal. 251(1), 1–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  67. Qing, J., Raske, D.: On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds. Int. Math. Res. Not. 20 (Art. ID 94172) (2006)

  68. Qing, J., Raske, D.: Compactness for conformal metrics with constant \(Q\) curvature on locally conformally flat manifolds. Calc. Var. Partial Differ. Equ. 26(3), 343–356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  69. Robert, F.: Positive solutions for a fourth order equation invariant under isometries. Proc. Am. Math. Soc. 131(5), 1423–1431 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  70. Robert, F.: Admissible \(Q\)-curvatures under isometries for the conformal GJMS operators. Nonlinear elliptic partial differential equations. Contemp. Math., 540, Am. Math. Soc. Providence, RI (2011)

  71. 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 

  72. Wei, J., Xu, X.: On conformal deformations of metrics on \({\mathbb{S}^n}\). J. Funct. Anal. 157, 292–325 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  73. Wei, J., Xu, X.: Prescribing \(Q\)-curvature problem on \({\mathbb{S}^n}\). J. Funct. Anal. 257, 1995–2023 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  74. Weinstein, G., Zhang, L.: The profile of bubbling solutions of a class of fourth order geometric equations on \(4\)-manifolds. J. Funct. Anal. 257(12), 3895–3929 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  75. Xu, X.: Uniqueness theorem for integral equations and its application. J. Funct. Anal. 247(1), 95–109 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  76. Yang, R.: On higher order extensions for the fractional Laplacian. arXiv:1302.4413

  77. Zhu, M.: Prescribing integral curvature equation. arXiv:1407.2967

Download references

Acknowledgments

Part of this work was done while J. Xiong was visiting Université Paris VII and Université Paris XII in November 2013 through a Sino-French research program in mathematics. He would like to thank Professors Yuxin Ge and Xiaonan Ma for the arrangement. He also thanks Professor Gang Tian for his support and encouragement. T. Jin would like to thank Professors Henri Berestycki and Luis Silvestre for their support and encouragement. The authors thank Professor Hongjie Dong for suggesting the current proof of Proposition 2.2. T. Jin was supported in part by NSF Grant DMS-1362525 and Hong Kong RGC Grant ECS 26300716. Y.Y. Li was supported in part by NSF Grants DMS-1203961 and NSF-DMS-1501004. J. Xiong was supported in part by NSFC 11501034, NSFC 11571019, Beijing MCESEDD (20131002701) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Tianling Jin

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    YanYan Li

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Jingang Xiong

Authors
  1. Tianling Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YanYan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jingang Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianling Jin.

Additional information

Communicated by F. C. Marques.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jin, T., Li, Y. & Xiong, J. The Nirenberg problem and its generalizations: a unified approach. Math. Ann. 369, 109–151 (2017). https://doi.org/10.1007/s00208-016-1477-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-016-1477-z

Mathematics Subject Classification

Search

Navigation