In this paper we study the solutions of Toda systems on Riemann surface in the critical case, proving a sufficient condition for existence.
@article{ASNSP_2005_5_4_4_703_0, author = {Li, Jiayu and Li, Yuxiang}, title = {Solutions for Toda systems on Riemann surfaces}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {703--728}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 4}, number = {4}, year = {2005}, zbl = {1170.35410}, mrnumber = {2207740}, language = {en}, url = {archive.numdam.org/item/ASNSP_2005_5_4_4_703_0/} }
Li, Jiayu; Li, Yuxiang. Solutions for Toda systems on Riemann surfaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 703-728. http://archive.numdam.org/item/ASNSP_2005_5_4_4_703_0/
[1] A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys. 143 (1992), 501-525. | MR 1145596 | Zbl 0745.76001
, , and ,[2] A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Part II, Commun. Math. Phys. 174 (1995), 229-260. | MR 1362165 | Zbl 0840.76002
, , and ,[3] Conformal deformation of metrics on , J. Differential Geom. 23 (1988), 259-296. | MR 925123 | Zbl 0649.53022
and ,[4] Prescribing Gaussian curvature on , Acta Math. 159 (1987), 214-259. | MR 908146 | Zbl 0636.53053
and ,[5] Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. | MR 1885666 | Zbl 1040.53046
and ,[6] Scalar curvature on , Trans. Amer. Math. Soc. 303 (1987), 365-382. | MR 896027 | Zbl 0635.35026
and ,[7] Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-622. | MR 1121147 | Zbl 0768.35025
and ,[8] The differential equation on a compact Riemann surface, Asian J. Math. 1 (1997), 230-248. | MR 1491984 | Zbl 0955.58010
, , and ,[9] An analysis of the two-vortex case in the Chern-Simons Higgs model, Calc. Var. Partial Differential Equations 7 (1998), 87-97. | MR 1624438 | Zbl 0928.58021
, , and ,[10] Multiplicity results for the two-vortex Chern-Simons Higgs model on the two sphere, Comment. Math. Helv. 74 (1999), 118-142. | MR 1677094 | Zbl 0913.53032
, , and ,[11] Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with order potentials, Commun. Math. Phys. 217 (2001), 383-407. | MR 1821229 | Zbl 0994.58009
, , , and ,[12] Existence of conformal metrics with constant -curvature, preprint, SISSA, 70/2004/M. | MR 2456884
and ,[13] Multivortex solutions of the Abelian Chern-Simons theory, Phys. Rev. Lett. 64 (1990), 2230-2233. | MR 1050529 | Zbl 1014.58500
, and ,[14] Self-dual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. | MR 1050530 | Zbl 1050.81595
and ,[15] Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., to appear. | MR 2199785 | Zbl pre05023695
, and ,[16] Analytic aspects of the Toda system. I. A Moser-Trudinger inequality, Comm. Pure Appl. Math. 54 (2001), 1289-1319. | MR 1846799 | Zbl 1099.35035
and ,[17] Curvature functions for compact -manifolds, Ann. of Math. 99 (1974), 14-47. | MR 343205 | Zbl 0273.53034
and ,[18] Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001), 163-192. | MR 1838044 | Zbl 0995.58021
,[19] The extremal functions for Moser-Trudinger inequality on compact Riemannian manifolds, China Ser. A 48 (2005), 618-648. | MR 2158479 | Zbl 1100.53036
,[20] Chern-Simons vortes theorey and Toda systems, J. Differential Equations 184 (2002), 443-474. | MR 1929885 | Zbl 1013.58009
and ,[21] Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. | MR 1710938 | Zbl 0951.58030
and ,[22] Vortex condensates for the Chern-Simons theory, Commun. Math. Phys. 213 (2000), 599-639. | MR 1785431 | Zbl 0998.81047
and ,[23] On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 109-121. | MR 1619043 | Zbl 0912.58046
and ,[24] Multiple condensate solutions for the Chern-Simons Higgs theory, J. Math. Phys. 37 (1996), 3769-3796. | MR 1400816 | Zbl 0863.58081
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