SpringerLink
Log in

A mean field type equation on vector bundles

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The mean field type equation is a basic problem of mathematics. Up to now, far too little attention has been paid to the mean field type equation on vector bundles. Recently, some existence results for a mean field type equation on line bundles over a closed Riemann surface were shown in our prior work. In the present paper, we generalize the previous results to vector bundles. Precisely, we prove the existence of minimizing solutions to a mean field type equation on vector bundles in the subcritical case by a direct method of calculus of variations and obtain a sufficient condition for the existence result in the critical case via the method of blow-up analysis. Taking account of the additional terms corresponding to the distributional decomposition of the bundle Laplace–Beltrami operator, we need delicate blow-up analysis and elaborate estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data availability is not applicable to this article as no new data were created or analyzed in this study.

References

  1. Aubin, T.: Sur la function exponentielle. C. R. Acad. Sci. Paris Sér. A-B 270, A1514–A1516 (1970)

    Google Scholar 

  2. Bartolucci, D., de Marchis, F., Malchiodi, A.: Supercritical conformal metrics on surfaces with conical singularities. Int. Math. Res Not. IMRN 24, 5625–5643 (2011)

    Article  MathSciNet  Google Scholar 

  3. Battaglia, L., López-Soriano, R.: A double mean field equation related to a curvature prescription problem. J. Differ. Equ. 269, 2705–2740 (2020)

    Article  MathSciNet  Google Scholar 

  4. Berger, M.: Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds. J. Differ. Geom. 5, 325–332 (1971)

    Article  MathSciNet  Google Scholar 

  5. Caffarelli, A., Yang, Y.S.: Vortex condensation in the Chern–Simons Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chang, K., Liu, J.: On Nirenberg’s problem. Internat. J. Math. 4(1), 35–58 (1993)

    Article  MathSciNet  Google Scholar 

  7. Chang, A., Yang, P.: Prescribing Gaussian curvature on \(S^2\). Acta Math. Sci. Ser. B 159, 215–259 (1987)

    Google Scholar 

  8. Chang, A., Yang, P.: Conformal deformation of metrics on \(S^2\). J. Differ. Geom. 23, 259–296 (1988)

    Google Scholar 

  9. Chen, W., Ding, W.: Scalar curvatures on \(S^2\). Trans. Am. Math. Soc. 303(1), 365–382 (1987)

    Google Scholar 

  10. Chen, W., Li, C.: Prescribing Gaussian curvatures on surfaces with conical singularities. J. Geom. Anal. 1, 359–372 (1991)

    Article  MathSciNet  Google Scholar 

  11. Chen, W., Li, C.: A priori estimate for the Nirenberg problem. Discrete Contin. Dyn. Syst. Ser. S 1(2), 225–233 (2008)

    MathSciNet  Google Scholar 

  12. Chen, C., Lin, C.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm. Pure Appl. Math. 55(6), 728–771 (2002)

    Article  MathSciNet  Google Scholar 

  13. Chen, C., Lin, C.: Topological degree for a mean field equation on Riemann surfaces. Commun. Pure Appl. Math. 56(12), 1667–1727 (2003)

    Article  MathSciNet  Google Scholar 

  14. D’Aprile, T., de Marchis, F., Ianni, I.: Prescribed Gauss curvature problem on singular surfaces. Calc. Var. Partial Differ. Equ. 57(4), 99 (2018)

    Article  MathSciNet  Google Scholar 

  15. de Marchis, F., López-Soriano, R.: Existence and non existence results for the singular Nirenberg problem. Calc. Var. Partial Differ. Equ. 55(2), 35 (2016)

    Article  MathSciNet  Google Scholar 

  16. de Marchis, F., López-Soriano, R., Ruiz, D.: Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials. J. Math. Pures Appl. 115, 237–267 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ding, W., Jost, J., Li, J., Wang, G.: The differential equation \(\Delta u = 8\pi -8\pi he^u\) on a compact Riemann Surface. Asian J. Math. 1, 230–248 (1997)

    Article  MathSciNet  Google Scholar 

  18. Ding, W., Jost, J., Li, J., Wang, G.: An analysis of the two-vortex case in the Chern–Simons Higgs model. Calc. Var. Partial Differ. Equ. 7, 87–97 (1998)

    Article  MathSciNet  Google Scholar 

  19. Ding, W., Jost, J., Li, J., Wang, G.: Multiplicity results for the two-vortex Chern–Simons Higgs model on the two-sphere. Comment. Math. Helv. 74, 118–142 (1999)

    Article  MathSciNet  Google Scholar 

  20. Ding, W., Jost, J., Li, J., Wang, G.: Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 16, 653–666 (1999)

    Article  MathSciNet  Google Scholar 

  21. Djadli, Z.: Existence result for the mean field problem on Riemann surfaces of all genuses. Commun. Contemp. Math. 10, 205–220 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Han, Z.: Prescribing Gaussian curvature on \(S^2\). Duke Math. J. 61(3), 679–703 (1990)

    Article  MathSciNet  Google Scholar 

  24. Hong, J., Kim, Y., Pac, P.: Multivortex solutions of the abelian Chern–Simons-Higgs theory. Phys. Rev. Lett. 64(19), 2230–2233 (1990)

    Article  MathSciNet  Google Scholar 

  25. Ji, M.: On positive scalar curvature on \(S^2\). Calc. Var. Partial Differ. Equ. 19(2), 165–182 (2004)

    Article  MathSciNet  Google Scholar 

  26. Kazdan, J., Warner, F.: Curvature functions for compact \(2\)-manifolds. Ann. Math. 99, 14–47 (1974)

    Article  MathSciNet  Google Scholar 

  27. Li, X.: An improved Trudinger–Moser inequality and its extremal functions involving \(L^p\)-norm in \({\mathbb{R} }^2\). Turk. J. Math. 44(4), 1092–1114 (2020)

    Article  Google Scholar 

  28. Li, Y.: Harnack type inequality: the method of moving planes. Commun. Math. Phys. 200(2), 421–444 (1999)

    Article  MathSciNet  Google Scholar 

  29. Li, Y., Liu, P., Yang, Y.: Moser–Trudinger inequalities on vector bundles over a compact Riemannian manifold of dimension 2. Calc. Var. Partial Differ. Equ. 28, 59–83 (2007)

    Article  MathSciNet  Google Scholar 

  30. Li, J., Sun, L., Yang, Y.: Boundary Value Problem for the Mean Field Equation on a Compact Riemann Surface (2022). ar**v:2201.01544

  31. Lin, C.: Topological degree for mean field equations on \(S^2\). Duke Math. J. 104(3), 501–536 (2000)

    Article  MathSciNet  Google Scholar 

  32. Malchiodi, A.: Topological methods for an elliptic equation with exponential nonlinearities. Discrete Contin. Dyn. Syst. 21, 277–294 (2008)

    Article  MathSciNet  Google Scholar 

  33. Malchiodi, A., Ruiz, D.: A variational analysis of the Toda system on compact surfaces. Comm. Pure Appl. Math. 66, 332–371 (2013)

    Article  MathSciNet  Google Scholar 

  34. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1091 (1971)

    Article  MathSciNet  Google Scholar 

  35. Moser, J.: On a nonlinear problem in differential geometry. In: Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador. pp. 273–280. Academic Press, New York, (1971)

  36. Nolasco, M.: Nontopological \(N\)-vortex condensates for the self-dual Chern–Simons theory. Commun. Pure Appl. Math. 56(12), 1752–1780 (2003)

    Article  MathSciNet  Google Scholar 

  37. Nolasco, M., Tarantello, G.: Double vortex condensates in the Chern–Simons–Higgs theory. Calc. Var. Partial Differ. Equ. 9(1), 31–94 (1999)

    Article  MathSciNet  Google Scholar 

  38. Peetre, J.: Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16(1), 279–317 (1966)

  39. Pohozaev, S.: The Sobolev embedding in the special case \(pl=n\). In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Sections, Moscov. Energet. Inst., Moscow , pp. 158–170 (1965)

  40. Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries. Acta Math. Sci. Ser. B 160, 19–64 (1988)

    MathSciNet  Google Scholar 

  41. Sun, L., Wang, Y., Yang, Y.: Existence Results for a Generalized Mean Field Equation on a Closed Riemann Surface. (2021). ar**v: 2101.03859

  42. Sun, L., Zhu, J.: Existence of Kazdan–Warner Equation with Sign-Changing Prescribed Function. (2021). ar**v: 2012.12840

  43. Tarantello, G.: Multiple condensate solutions for the Chern–Simons–Higgs theory. J. Math. Phys. 37, 3769–3796 (1996)

    Article  MathSciNet  Google Scholar 

  44. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MathSciNet  Google Scholar 

  45. Wang, M., Liu, Q.: The equation \(\Delta u+\nabla \phi \cdot \nabla u=8 \pi c\left(1-h e^{u}\right)\) on a Riemann surface. J. Partial Differ. Equ. 25, 335–355 (2012)

    MathSciNet  Google Scholar 

  46. Wang, Y., Yang, Y.: A mean field type flow with sign-changing prescribed function on a symmetric Riemann surface. J. Funct. Anal. 282(109449), 31 (2022)

    MathSciNet  Google Scholar 

  47. Yang, J.: A weighted Trudinger–Moser inequality on a closed Riemann surface with a finite isometric group action. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116(2), 18 (2022)

  48. Yang, J., Yang, Y.: Mean Field Type Equations on Line Bundle Over a Closed Riemann Surface. (2022). ar**v:2206.01437

  49. Yang, J., Yang, Y.: Extremal sections for a Trudinger–Moser functional on vector bundle over a closed Riemann surface. J. Geom. Anal. 34(2), 41 (2024)

    Article  MathSciNet  Google Scholar 

  50. Yang, Y.: Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two. J. Differ. Equ. 258(9), 3161–3193 (2015)

    Article  MathSciNet  Google Scholar 

  51. Yang, Y., Zhu, X.: A remark on a result of Ding–Jost–Li–Wang. Proc. Am. Math. Soc. 145, 3953–3959 (2017)

    Article  MathSciNet  Google Scholar 

  52. Yang, Y., Zhu, X.: Existence of solutions to a class of Kazdan–Warner equations on compact Riemannian surface. Sci. China Math. 61, 1109–1128 (2018)

    Article  MathSciNet  Google Scholar 

  53. Yang, Y., Zhu, X.: Mean field equations on a closed Riemannian surface with the action of an isometric group. Internat. J. Math. 31(2050072), 26 (2020)

    MathSciNet  Google Scholar 

  54. Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Docl. 2, 746–749 (1961)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Renmin University of China, Bei**g, 100872, China

    Jie Yang

Authors
  1. Jie Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jie Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J. A mean field type equation on vector bundles. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 128 (2024). https://doi.org/10.1007/s13398-024-01622-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-024-01622-y

Keywords

Mathematics Subject Classification

Search

Navigation