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.

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.

Classification: 35J60, 58E11
Chen, Chiun-Chuan 1; Lin, Chang-Shou 2; Wang, Guofang 3

1 Department of Mathematics Taiwan University Taipei, Taiwan
2 Department of Mathematics Chung-Cheng University Minghsiung, Chia-Yi 621 Taiwan
3 Max-Planck-Institute for Mathematics in the Sciences Inselstr. 22-26 04103 Leipzig, Germany
@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},
     pages = {367--397},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {2},
     year = {2004},
     mrnumber = {2075988},
     zbl = {1170.35413},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0/}
}
TY  - JOUR
AU  - Chen, Chiun-Chuan
AU  - Lin, Chang-Shou
AU  - Wang, Guofang
TI  - Concentration phenomena of two-vortex solutions in a Chern-Simons model
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 367
EP  - 397
VL  - 3
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0/
LA  - en
ID  - ASNSP_2004_5_3_2_367_0
ER  - 
%0 Journal Article
%A Chen, Chiun-Chuan
%A Lin, Chang-Shou
%A Wang, Guofang
%T Concentration phenomena of two-vortex solutions in a Chern-Simons model
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 367-397
%V 3
%N 2
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0/
%G en
%F 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/

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[15] W. Ding - J. Jost - J. Li - G. Wang, The differential equation of Δu=8π-8πhe u on a compact Riemann surface, Asian J. Math., 1 (1997), 230-248. | MR | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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