Solutions for Toda systems on Riemann surfaces

Li, Jiayu; Li, Yuxiang

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 703-728.

In this paper we study the solutions of Toda systems on Riemann surface in the critical case, proving a sufficient condition for existence.

Classification : 35J60, 58G03, 35J45

[1] E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys. 143 (1992), 501-525. | MR | Zbl

[2] E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, 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 | Zbl

[3] A. S. Y. Chang and P. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom. 23 (1988), 259-296. | MR | Zbl

[4] A. S. Y. Chang and P. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math. 159 (1987), 214-259. | MR | Zbl

[5] C.-C. Chen and C.-S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. | MR | Zbl

[6] W. X. Chen and W. Y. Ding, Scalar curvature on $S^2$, Trans. Amer. Math. Soc. 303 (1987), 365-382. | MR | Zbl

[7] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-622. | MR | Zbl

[8] W. Ding, J. Jost, J. Li and G. Wang, The differential equation $\Delta u=8\pi -8\pi h{e}^u$ on a compact Riemann surface, Asian J. Math. 1 (1997), 230-248. | MR | Zbl

[9] W. Ding, J. Jost, J. Li and G. Wang,An analysis of the two-vortex case in the Chern-Simons Higgs model, Calc. Var. Partial Differential Equations 7 (1998), 87-97. | MR | Zbl

[10] W. Ding, J. Jost, J. Li and G. Wang, Multiplicity results for the two-vortex Chern-Simons Higgs model on the two sphere, Comment. Math. Helv. 74 (1999), 118-142. | MR | Zbl

[11] W. Ding, J. Jost, J. Li, X. Peng and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with $6^{th}$ order potentials, Commun. Math. Phys. 217 (2001), 383-407. | MR | Zbl

[12] Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant $Q$-curvature, preprint, SISSA, 70/2004/M. | MR

[13] J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the Abelian Chern-Simons theory, Phys. Rev. Lett. 64 (1990), 2230-2233. | MR | Zbl

[14] R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. | MR | Zbl

[15] J. Jost, C.-S. Lin and G. Wang, Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., to appear. | MR

[16] J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality, Comm. Pure Appl. Math. 54 (2001), 1289-1319. | MR | Zbl

[17] J. Kazdan and F. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. 99 (1974), 14-47. | MR | Zbl

[18] Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001), 163-192. | MR | Zbl

[19] Y. Li, The extremal functions for Moser-Trudinger inequality on compact Riemannian manifolds, China Ser. A 48 (2005), 618-648. | MR | Zbl

[20] M. Lucia and M. Nolasco, Chern-Simons vortes theorey and Toda systems, J. Differential Equations 184 (2002), 443-474. | MR | Zbl

[21] M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. | MR | Zbl

[22] M. Nolasco and G. Tarantello, Vortex condensates for the $\mathrm{SU}\left(3\right)$ Chern-Simons theory, Commun. Math. Phys. 213 (2000), 599-639. | MR | Zbl

[23] M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 109-121. | MR | Zbl

[24] G. Tarantello, Multiple condensate solutions for the Chern-Simons Higgs theory, J. Math. Phys. 37 (1996), 3769-3796. | MR | Zbl