On condensate of solutions for the Chern–Simons–Higgs equation

Lin, Chang-Shou; Yan, Shusen

doi:10.1016/j.anihpc.2016.10.006

Lin, Chang-Shou ; Yan, Shusen

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1329-1354.

Résumé

This is the first part of our comprehensive study on the structure of doubly periodic solutions for the Chern–Simons–Higgs equation with a small coupling constant. We first classify the bubbling type of the blow-up point according to the limit equations. Assuming that all the blow-up points are away from the vortex points, we prove the non-coexistence of different bubbling types in a sequence of bubbling solutions. Secondly, for the CS type bubbling solutions, we obtain an existence result without the condition on the blow-up set as in [4]. This seems to be the first general existence result of the multi-bubbling CS type solutions which is obtained under nearly necessary conditions. Necessary and sufficient conditions are also discussed for the existence of bubbling solutions blowing up at vortex points.

MR Zbl

DOI : 10.1016/j.anihpc.2016.10.006

Mots clés : Condensate, Chern–Simons–Higgs equation, Bubbling phenomena, Pohozaev identity, Uniqueness

@article{AIHPC_2017__34_5_1329_0,
     author = {Lin, Chang-Shou and Yan, Shusen},
     title = {On condensate of solutions for the {Chern{\textendash}Simons{\textendash}Higgs} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1329--1354},
     publisher = {Elsevier},
     volume = {34},
     number = {5},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.10.006},
     zbl = {1387.35512},
     mrnumber = {3742527},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2016.10.006/}
}

TY  - JOUR
AU  - Lin, Chang-Shou
AU  - Yan, Shusen
TI  - On condensate of solutions for the Chern–Simons–Higgs equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 1329
EP  - 1354
VL  - 34
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2016.10.006/
DO  - 10.1016/j.anihpc.2016.10.006
LA  - en
ID  - AIHPC_2017__34_5_1329_0
ER  -

%0 Journal Article
%A Lin, Chang-Shou
%A Yan, Shusen
%T On condensate of solutions for the Chern–Simons–Higgs equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 1329-1354
%V 34
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2016.10.006/
%R 10.1016/j.anihpc.2016.10.006
%G en
%F AIHPC_2017__34_5_1329_0

Lin, Chang-Shou; Yan, Shusen. On condensate of solutions for the Chern–Simons–Higgs equation. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1329-1354. doi : 10.1016/j.anihpc.2016.10.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2016.10.006/

Bibliographie
Cité par

[1] W. Ao, C.-S. Lin, J. Wei, On the non-topological solutions of the $A_{2}$ and $B_{2}$ Chern–Simons system, preprint. | MR

[2] Bartolucci, D.; Lee, Y.; Lin, C.-S.; Onodera, M. Asymptotic analysis of solutions to a gauged $O (3)$ sigma model, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015) no. 3, pp. 651–685 | DOI | Numdam | MR | Zbl

[3] Caffarelli, L.; Yang, Y. Vortex condensation in the Chern–Simons Higgs model: an existence theorem, Commun. Math. Phys., Volume 168 (1995) no. 2, pp. 321–336 | DOI | MR | Zbl

[4] Chan, H.; Fu, C.-C.; Lin, C.-S. Non-topological multi-vortex solutions to the self-dual Chern–Simons–Higgs equation, Commun. Math. Phys., Volume 231 (2002) no. 2, pp. 189–221 | DOI | MR | Zbl

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

[6] Chen, C.-C.; Lin, C.-S. Topological degree for a mean field equation on Riemann surfaces, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1667–1727 | MR | Zbl

[7] Chen, S.; Yang, Y. Existence of multiple vortices in supersymmetric gauge field theory, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., Volume 468 (2012) no. 2148, pp. 3923–3946 | MR | Zbl

[8] Chern, J.-L.; Chen, Z.-Y.; Lin, C.-S. Uniqueness of topological solutions and the structure of solutions for the Chern–Simons system with two Higgs particles, Commun. Math. Phys., Volume 296 (2010), pp. 323–351 | MR | Zbl

[9] Choe, K. Asymptotic behavior of condensate solutions in the Chern–Simons–Higgs theory, J. Math. Phys., Volume 48 (2007), pp. 103501 | DOI | MR | Zbl

[10] Choe, K.; Kim, N. Blow-up solutions of the self-dual Chern–Simons–Higgs vortex equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 25 (2008), pp. 313–338 | DOI | Numdam | MR | Zbl

[11] Dunne, G.V. Aspects topologiques de la physique en basse dimension, Topological Aspects of Low Dimensional Systems, EDP Sci., Les Ulis (1999), pp. 177–263 (Les Houches, 1998) | MR | Zbl

[12] Gudnason, S.B. Non-Abelian Chern–Simons vertices with generic groups, Nucl. Phys. B (2009), pp. 151–169 | DOI | MR | Zbl

[13] Gudnason, S.B. Fractional and semi-local non-Abelian Chern–Simons vortices, Nucl. Phys. B, Volume 840 (2010), pp. 160–185 | DOI | MR | Zbl

[14] Han, X.; Lin, C.-S. Existence of non-Abelian vortices with product gauge groups, Nucl. Phys. B, Volume 878 (2014), pp. 117–149 | MR | Zbl

[15] X. Han, C.-S. Lin, G. Tarantello, Y. Yang, Chern–Simons vertices in the Gudnason model, preprint. | MR | Zbl

[16] Han, X.; Tarantello, G. Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model, Calc. Var. Partial Differ. Equ., Volume 49 (2014), pp. 1149–1176 | MR | Zbl

[17] Hong, J.; Kim, Y.; Pac, P. Multivortex solutions of the abelian Chern–Simons–Higgs theory, Phys. Rev. Lett., Volume 64 (1990), pp. 2230–2233 | DOI | MR | Zbl

[18] Huang, H.; Lin, C.-S. On the entire radial solutions of the Chern–Simons SU(3) system, Commun. Math. Phys., Volume 327 (2014) no. 3, pp. 815–848 | DOI | MR | Zbl

[19] Jackiw, R.; Weinberg, E. Self-dual Chern–Simons vortex, Phys. Rev. Lett., Volume 64 (1990), pp. 2234–2237 | DOI | MR | Zbl

[20] Jaffe, A.; Taubes, C. Vortices and Monopoles, Prog. Phys., vol. 2, Birkhäuser Boston, Boston, MA, 1990

[21] Li, Yan Yan Harnack type inequality: the method of moving planes, Commun. Math. Phys., Volume 200 (1999), pp. 421–444 | MR | Zbl

[22] Lin, C.-S.; Ponce, C.; Yang, Y. A system of elliptic equations arising in Chern–Simons field theory, J. Funct. Anal., Volume 247 (2007) no. 2, pp. 289–350 | MR | Zbl

[23] Lin, C.-S.; Prajapat, J.V. Vortex condensates for relativistic Abelian Chern–Simons model with two Higgs scalar fields and two gauge fields on a torus, Commun. Math. Phys., Volume 288 (2009), pp. 311–347 | MR | Zbl

[24] Lin, C.-S.; Yan, S. Bubbling solutions for relativistic Abelian Chern–Simons model on a torus, Commun. Math. Phys., Volume 297 (2010) no. 3, pp. 733–758 | MR | Zbl

[25] Lin, C.-S.; Yan, S. Existence of bubbling solutions for Chern–Simons model on a torus, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 2, pp. 353–392 | MR | Zbl

[26] Lin, C.-S.; Yan, S. Bubbling solutions for the SU(3) Chern–Simons model on a torus, Commun. Pure Appl. Math., Volume 66 (2013), pp. 991–1027 | MR | Zbl

[27] Lin, C.-S.; Yang, Y. Non-Abelian multiple vortices in supersymmetric field theory, Commun. Math. Phys., Volume 304 (2011), pp. 433–457 | MR | Zbl

[28] Lin, C.-S.; Yang, Y. Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions, Nucl. Phys. B, Volume 846 (2011) no. 3, pp. 650–676 | MR | Zbl

[29] Lozano, G.; Marqués, D.; Moreno, E.; Schaposnik, F. Non-Abelian Chern–Simons vortices, Phys. Lett. B, Volume 654 (2007), pp. 27–34 | DOI | MR | Zbl

[30] Malchiodi, A.; Ndiaye, C. Some existence results for the Toda system on closed surfaces, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. Rend. Lincei (9) Mat. Appl., Volume 18 (2007) no. 4, pp. 391–412 | MR | Zbl

[31] Nolasco, M.; Tarantello, G. Double vortex condensates in the Chern–Simons–Higgs theory, Calc. Var. Partial Differ. Equ., Volume 9 (1999), pp. 31–94 | DOI | MR | Zbl

[32] Nolasco, M.; Tarantello, G. Vortex condensates for the $SU (3)$ Chern–Simons theory, Commun. Math. Phys., Volume 213 (2000) no. 3, pp. 599–639 | DOI | MR | Zbl

[33] Poliakovsky, A.; Tarantello, G. On a planar Liouville-type problem in the study of selfgravitating strings, J. Differ. Equ., Volume 252 (2012), pp. 3668–3693 | DOI | MR | Zbl

[34] Prajapat, J.; Tarantello, G. On a class of elliptic problems in $R^{2}$ : symmetry and uniqueness results, Proc. R. Soc. Edinb., Sect. A, Volume 131 (2001) no. 4, pp. 967–985 | DOI | MR | Zbl

[35] Tarantello, G. Uniqueness of selfdual periodic Chern–Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., Volume 29 (2007) no. 2, pp. 191–217 | DOI | MR | Zbl

[36] Tarantello, G. Self-Dual Gauge Field Vortices: An Analytical Approach, Springer, 2007 | MR

[37] Yang, Y. The relativistic non-abelian Chern–Simons equations, Commun. Math. Phys., Volume 186 (1997), pp. 199–218 | DOI | MR | Zbl

[38] Yang, Y. Solitons in Field Theory and Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, New York, 2001 | DOI | MR | Zbl

Cité par Sources :