Concentration phenomena of two-vortex solutions in a Chern-Simons model

Chen, Chiun-Chuan; Lin, Chang-Shou; Wang, Guofang

Chen, Chiun-Chuan; Lin, Chang-Shou; Wang, Guofang

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 2, p. 367-397

Abstract

By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.

MR 2075988 | Zbl 1170.35413 | 2 citations in Numdam

Classification: 35J60, 58E11

@article{ASNSP_2004_5_3_2_367_0,
     author = {Chen, Chiun-Chuan and Lin, Chang-Shou and Wang, Guofang},
     title = {Concentration phenomena of two-vortex solutions in a Chern-Simons model},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {2},
     year = {2004},
     pages = {367-397},
     zbl = {1170.35413},
     mrnumber = {2075988},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0}
}

Chen, Chiun-Chuan; Lin, Chang-Shou; Wang, Guofang. Concentration phenomena of two-vortex solutions in a Chern-Simons model. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 2, pp. 367-397. http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0/

References

[1] L. Ahlfors, “Complex analysis”, 2nd edition, McGraw-Hill Book Co., New York, 1966. | MR 510197 | Zbl 0395.30001

[2] D. Bartolucci - G. Tarantello, Liouville type equations with singular data and their application to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), 3-47. | MR 1917672 | Zbl 1009.58011

[3] H. Brezis - F. Merle, Uniform estimates and blow-up behavior for solutions of $- Δ u = V (x) e^{u}$ in two dimensions, Comm. Partial Differential Equation 16 (1991), 1223-1253. | MR 1132783 | Zbl 0746.35006

[4] R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 321-347. | MR 955072 | Zbl 0635.53047

[5] L. Caffarelli - Y. Yang, Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys. 168 (1995), 321-336. | MR 1324400 | Zbl 0846.58063

[6] E. Caglioti - P. L. Lions - C. Marchioro - M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501-525. | MR 1145596 | Zbl 0745.76001

[7] H. Chan - C. C. Fu - C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equations, Comm. Math. Phys. 231 (2002), 189-221. | MR 1946331 | Zbl 1018.58008

[8] S. Chanillo - M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217-238. | MR 1262195 | Zbl 0821.35044

[9] S. Y. Chang - P. Yang, Prescribing Gaussian curvature on $S^{2}$ , Acta Math. 159 (1987), 215-259. | MR 908146 | Zbl 0636.53053

[10] C. C. Chen - C. S. Lin, On the symmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 271-296. | Numdam | MR 1831657 | Zbl 0995.35004

[11] C. C. Chen - C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 4 (2002), 728-771. | MR 1885666 | Zbl 1040.53046

[12] C. C. Chen and C. S. Lin, Topological Degree for a Mean Field Equation on Riemann Surfaces, Comm. Pure Appl. Math. 56 (2003), 1667-1707. | MR 2001443 | Zbl 1032.58010

[13] K. S. Chou - Tom Y. H. Wan, Asymptotic radial symmetry for solutions of $Δ u + e^{u} = 0$ in a punctured disc, Pacific J. Math. 163 (1994), 269-276. | MR 1262297 | Zbl 0794.35049

[14] W. Ding - J. Jost - J. Li - G. Wang, Multiplicity results of two-vortex Chern-Simons-Higgs model on the two-sphere, Comm. Math. Helv. 74 (1999), 118-142. | MR 1677094 | Zbl 0913.53032

[15] W. Ding - J. Jost - J. Li - G. Wang, The differential equation of $Δ u = 8 π - 8 π h e^{u}$ on a compact Riemann surface, Asian J. Math., 1 (1997), 230-248. | MR 1491984 | Zbl 0955.58010

[16] W. Ding - J. Jost - J. Li - X. Peng - G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functional with 6th order potenliatls, Comm. Math. Phys. 217 (2001), 383-407. | MR 1821229 | Zbl 0994.58009

[17] G. Dunne, “Self-dual Chern-Simons Theories”, Lecture Notes in Physics m36, Springer-Verlag, Berlin, 1995. | Zbl 0834.58001

[18] J. Hong - Y. Kim - P. Y. Pac, Multivortex solutions of the Abelian Chern Simons theory, Phys. Rev. Letter 64 (1990), 2230-2233. | MR 1050529 | Zbl 1014.58500

[19] R. Jackiw - E. J. Weinberg, Selfdual Chern Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. | MR 1050530 | Zbl 1050.81595

[20] Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), 421-444. | MR 1673972 | Zbl 0928.35057

[21] C. S. Lin, Topological degree for mean field equations on $S^{2}$ , Duke Math. J. 104 (2000), 501-536. | MR 1781481 | Zbl 0964.35038

[22] C. S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal. 153 (2000), 153-176. | MR 1770683 | Zbl 0968.35045

[23] M. Nolasco, Non-topological $N$ -vortex condensates for the self-dual chern-Simons theory, Comm. Pure Appl. Math. 56 (2003), 1752-1780. | MR 2001445 | Zbl 1032.58005

[24] M. Nolasco - G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. | MR 1710938 | Zbl 0951.58030

[25] M. Nolasco - G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal. 145 (1998), 161-195. | MR 1664542 | Zbl 0980.46022

[26] J. Prajapat - G. Tarantello, On a class of elliptic problems in $ℝ^{2}$ : symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 967-985. | MR 1855007 | Zbl 1009.35018

[27] Spruck - Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 75-97. | Numdam | MR 1320569 | Zbl 0836.35007

[28] Taubes, Arbitrary $N$ -vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277-292. | MR 573986 | Zbl 0451.35101