Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 539–562 | Cite as

Variational analysis of Toda systems

  • Andrea Malchiodi


The author surveys some recent progress on the Toda system on a twodimensional surface Σ, arising in models from self-dual non-abelian Chern-Simons vortices, as well as in differential geometry. In particular, its variational structure is analysed, and the role of the topological join of the barycentric sets of Σ is shown.


Geometric PDEs Variational Methods Min-max Schemes 

2000 MR Subject Classification

35B33 35J35 53A30 53C21 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aubin, T., Some Nonlinear Problems in Differential Geometry, Springer-Verlag, New York, 1998.CrossRefMATHGoogle Scholar
  2. [2]
    Bahri, T. and Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41, 1988, 253–294.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Bartolucci, D., de Marchis, F. and Malchiodi, A., Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not., 2011(24), 2011, 5625–5643.MathSciNetMATHGoogle Scholar
  4. [4]
    Bartolucci, D. and Malchiodi, A., An improved geometric inequality via vanishing moments, with applications to singular Liouville equations, Comm. Math. Phys., 322(2), 2013, 415–452.MATHGoogle Scholar
  5. [5]
    Battaglia, L., Existence and multiplicity result for the singular Toda system, J. Math. Anal. Appl., 424(1), 2015, 49–85.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Battaglia, L. and Mancini, G., A note on compactness properties of the singular Toda system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26(3), 2015, 299–307.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Battaglia, L., Jevnikar, A., Malchiodi, A. and Ruiz, D., A general existence result for the Toda system on compact surfaces, Adv. in Math., 285, 2015, 937–979.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Bolton, J. and Woodward, L. M., Some geometrical aspects of the 2-dimensional Toda equations (in Geometry), Topology and Physics, Campinas, 1996, 69–81; de Gruyter, Berlin, 1997.MATHGoogle Scholar
  9. [9]
    Brezis, H. and Merle, F., Uniform estimates and blow-up behavior for solutions of u = V (x)eu in two dimensions, Commun. Partial Differ. Equations, 16(8–9), 1991, 1223–1253.CrossRefMATHGoogle Scholar
  10. [10]
    Calabi, E., Isometric imbedding of complex manifolds, Ann. Math., 58(2), 1953, 1–23.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Carlotto, A. and Malchiodi, A., Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262(2), 2012, 409–450.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Chanillo, S. and Kiessling, M., Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys., 160, 1994, 217–238.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Chen, W. X. and Li, C., Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal., 1(4), 1991, 359–372.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Chen, C. C. and Lin, C. S., Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56(12), 2003, 1667–1727.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Chen, X. X., Remarks on the existence of branch bubbles on the blowup analysis of equation -u = e2u in dimension two, Commun. Anal. Geom., 7(2), 1999, 295–302.CrossRefMATHGoogle Scholar
  16. [16]
    Chern, S. S. and Wolfson, J. G., Harmonic maps of the two-sphere into a complex Grassmann manifold,II, Ann. of Math., 125(2), 1987, 301–335.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Daprile, T., Pistoia, A. and Ruiz, D., Asymmetric blow-up for the SU(3) Toda system, J. Funct. Anal., 271(3), 2016, 495–531.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Ding, W., Jost, J., Li, J. and Wang, G., The differential equation u = 8p-8pheu on a compact Riemann surface, Asian J. Math., 1, 1997, 230–248.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Ding, W., Jost, J., Li, J. and Wang, G., Existence results for mean field equations, Ann. Inst. Henri Poincaré, Anal. Non Linèaire, 16(5), 1999, 653–666.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Djadli, Z., Existence result for the mean field problem on Riemann surfaces of all genus, Comm. Contemp. Math., 10(2), 2008, 205–220.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Djadli, Z. and Malchiodi, A., Existence of conformal metrics with constant Q-curvature, Ann. Math., 168(3), 2008, 813–858.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Dunne, G., Self-dual Chern-Simons Theories, Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin, 1995.CrossRefMATHGoogle Scholar
  23. [23]
    Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002.MATHGoogle Scholar
  24. [24]
    Jevnikar, A., Kallel, S. and Malchiodi, A., A topological join construction and the Toda system on compact surfaces of arbitrary genus, Anal. PDE, 8(8), 2015, 1963–2027.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Jost, J., Lin, C. S. and Wang, G., Analytic aspects of the Toda system II,Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., 59, 2006, 526–558.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Jost, J. and Wang, G., Analytic aspects of the Toda system I, AMoser-Trudinger inequality, Comm. Pure Appl. Math., 54, 2001, 1289–1319.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Kallel, S. and Karoui, R., Symmetric joins and weighted barycenters, Advanced Nonlinear Studies, 11, 2011, 117–143.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Li, J. and Li, Y., Solutions for Toda systems on Riemann surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 5), 4(4), 2005, 703–728.MathSciNetMATHGoogle Scholar
  29. [29]
    Li, Y. Y., Harnack type inequality: The method of moving planes, Commun. Math. Phys., 200(2), 1999, 421–444.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    Li, Y. Y. and Shafrir, I., Blow-up analysis for solutions of -?u = V eu in dimension two, Indiana Univ. Math. J., 43(4), 1994, 1255–1270.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Lin, C. S., Wei, J. and Yang, W., Degree counting and shadow system for SU(3) Toda system: One bubbling, 2014, preprint. Scholar
  32. [32]
    Lin, C. S., Wei, J. and Zhao, C., Sharp estimates for fully bubbling solutions of a SU(3) Toda system, Geom. Funct. Anal., 22(6), 2012, 1591–1635.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    Lin, C. S. and Zhang, L., A topological degree counting for some Liouville systems of mean field type, Comm. Pure Appl. Math., 64, 2011, 556–590.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Lucia, M., A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30(1), 2007, 113–138.MathSciNetMATHGoogle Scholar
  35. [35]
    Malchiodi, A., Morse theory and a scalar field equation on compact surfaces, Adv. Diff. Eq., 13, 2008, 1109–1129.MathSciNetMATHGoogle Scholar
  36. [36]
    Malchiodi, A. and Ndiaye, C. B., Some existence results for the Toda system on closed surfaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18(4), 2007, 391–412.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    Malchiodi, A. and Ruiz, D., New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geom. Funct. Anal., 21(5), 2011, 1196–1217.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    Malchiodi, A. and Ruiz, D., A variational analysis of the Toda system on compact surfaces, Comm. Pure Appl. Math., 66(3), 2013, 332–371.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    Malchiodi, A. and Ruiz, D., On the Leray-Schauder degree of the Toda system on compact surfaces, Proc. Amer. Math. Soc., 143(7), 2015, 2985–2990.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    Nolasco, M. and Tarantello, G., On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal., 145, 1998, 161–195.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    Prajapat, J. and Tarantello, G., On a class of elliptic problems in R2: Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh, Ser. A, 131, 2001, 967–985.CrossRefMATHGoogle Scholar
  42. [42]
    Struwe, M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv., 60, 1985, 558–581.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    Struwe, M. and Tarantello, G., On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat., 8(1), 1998, 109–121.MathSciNetMATHGoogle Scholar
  44. [44]
    Tarantello, G., Self-Dual Gauge Field Vortices: An Analytical Approach, PNLDE 72, Birkhäuser Boston, Boston, MA,2007.MATHGoogle Scholar
  45. [45]
    Yang, Y., Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, New York, 2001.CrossRefMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

Personalised recommendations