Skip to main content
SpringerLink
Log in

Onofri-Type Inequalities for Singular Liouville Equations

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities.

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

Similar content being viewed by others

References

  1. Bartolucci, D., Tarantello, G.: Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229(1), 3–47 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartolucci, D., Chen, C.-C., Lin, C.-S., Tarantello, G.: Profile of blow-up solutions to mean field equations with singular data. Commun. Partial Differ. Equ. 29(7–8), 1241–1265 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Bartolucci, D., Malchiodi, A.: An improved geometric inequality via vanishing moments, with applications to singular Liouville equations. Commun. Math. Phys. 322(2), 415–452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartolucci, D., Montefusco, E.: Blow-up analysis, existence and qualitative properties of solutions for the two-dimensional Emden-Fowler equation with singular potential. Math. Methods Appl. Sci. 30(18), 2309–2327 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bolton, J., Woodward, L.M.: Some geometrical aspects of the \(2\)-dimensional Toda equations. In: Geometry, Topology and Physics (Campinas, 1996), pp. 69–81. de Gruyter, Berlin (1997)

  7. Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^u\) in two dimensions. Commun. Partial Differ. Equ. 16(8–9), 1223–1253 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. 2(58), 1–23 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlotto, A.: On the solvability of singular Liouville equations on compact surfaces of arbitrary genus. Trans. Am. Math. Soc. 366(3), 1237–1256 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlotto, A., Malchiodi, A.: A class of existence results for the singular Liouville equation. C. R. Math. Acad. Sci. Paris 349(3–4), 161–166 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carlotto, A., Malchiodi, A.: Weighted barycentric sets and singular Liouville equations on compact surfaces. J. Funct. Anal. 262(2), 409–450 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, S.-Y.A., Yang, P.C.: Conformal deformation of metrics on \(S^2\). J. Differ. Geom. 27(2), 259–296 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Chng, S.-Y.A., Yang, P.C.: Prescribing Gaussian curvature on \(S^2\). Acta Math. 159(3–4), 215–259 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, W.X., Li, C.: Prescribing Gaussian curvatures on surfaces with conical singularities. J. Geom. Anal. 1(4), 359–372 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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(2), 230–248 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helv. 68(3), 415–454 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Y.Y., Shafrir, I.: Blow-up analysis for solutions of \(-\Delta u=Ve^u\) in dimension two. Indiana Univ. Math. J. 43(4), 1255–1270 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Malchiodi, A., Ruiz, D.: New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces. Geom. Funct. Anal. 21(5), 1196–1217 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moser, J.: A sharp form of an inequality. N. Trudinger (eds), Indiana Univ. Math. J., 20, 1077–1092 (1970/1971)

  23. Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86(3), 321–326 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Prajapat, J., Tarantello, G.: On a class of elliptic problems in \({\mathbb{R}}^2\): symmetry and uniqueness results. Proc. R. Soc. Edinburgh Sect. A 131(4), 967–985 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gabriella, T.: Selfdual gauge field vortices: an analytical approach. Progress in Nonlinear Differential Equations and their Applications, 72. Birkhäuser Boston Inc, Boston (2008)

  26. Tarantello, G.: Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete Contin. Dyn. Syst. 28(3), 931–973 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324(2), 793–821 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to express his gratitude to Professor Andrea Malchiodi for many valuable discussions and for his guidance during the preparation of this work. The author is supported by the FIRB Project Analysis and Beyond, by the PRINs Variational Methods and Nonlinear PDE’s and Variational and perturbative aspects of nonlinear differential problems and by the Mathematics Department at the University of Warwick.

Author information

Authors and Affiliations

  1. S.I.S.S.A/I.S.A.S, Via Bonomea 265, 34136, Trieste, Italy

    Gabriele Mancini

Authors
  1. Gabriele Mancini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriele Mancini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mancini, G. Onofri-Type Inequalities for Singular Liouville Equations. J Geom Anal 26, 1202–1230 (2016). https://doi.org/10.1007/s12220-015-9589-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-015-9589-3

Keywords

Mathematics Subject Classification

Search

Navigation