Asymptotic analysis of solutions to a gauged O(3) sigma model
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 3, pp. 651-685.

We analyze an elliptic equation arising in the study of the gauged O(3) sigma model with the Chern–Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in 2 , which implies the possibility that a stable solution which blows up at a vortex point exists. To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

DOI: 10.1016/j.anihpc.2014.03.001
Keywords: Gauged O(3) sigma models, Blow up analysis, Pohozaev type identity, Stable solutions
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     title = {Asymptotic analysis of solutions to a gauged $ \mathrm{O}(3)$ sigma model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Bartolucci, Daniele; Lee, Youngae; Lin, Chang-Shou; Onodera, Michiaki. Asymptotic analysis of solutions to a gauged $ \mathrm{O}(3)$ sigma model. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 3, pp. 651-685. doi : 10.1016/j.anihpc.2014.03.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.03.001/

[1] K. Arthur, D. Tchrakian, Y. Yang, Topological and nontopological self-dual Chern–Simons solitons in a gauged O(3) model, Phys. Rev. D 54 (1996), 5245 -5258 | MR

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

[3] A.A. Belavin, A.M. Polyakov, Metastable states of two dimensional isotropic ferromagnets, JETP Lett. 22 (1975), 245 -247

[4] D. Chae, H.-S. Nam, Multiple existence of the multivortex solutions of the self-dual Chern–Simons CP(1) model on a doubly periodic domain, Lett. Math. Phys. 49 (1999), 297 -315 | MR | Zbl

[5] K.-S. Cheng, C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in 2 , Math. Ann. 308 (1997), 119 -139 | MR | Zbl

[6] K. Choe, Existence of nontopological solutions in the Chern–Simons gauged O(3) sigma models, preprint.

[7] K. Choe, J. Han, Existence and properties of radial solutions in the self-dual Chern–Simons O(3) sigma model, J. Math. Phys. 52 (2011), 082301 | MR | Zbl

[8] K. Choe, J. Han, C.-S. Lin, T.-C. Lin, Uniqueness and solution structure of nonlinear equations arising from the Chern–Simons gauged O(3) sigma models, J. Differ. Equ. 255 (2013), 2136 -2166 | MR | Zbl

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

[10] K. Choe, N. Kim, C.-S. Lin, Existence of self-dual non-topological solutions in the Chern–Simons Higgs model, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 (2011), 837 -852 | Numdam | MR | Zbl

[11] K. Choe, H.-S. Nam, Existence and uniqueness of topological multivortex solutions of the self-dual Chern–Simons CP(1) model, Nonlinear Anal. 66 (2007), 2794 -2813 | MR | Zbl

[12] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equation in n , Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud. vol. 7A , Academic Press, New York–London (1981), 369 -402

[13] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer, Berlin (1983) | MR | Zbl

[14] P.K. Ghosh, S.K. Ghosh, Topological and nontopological solitons in a gauged O(3) sigma model with Chern–Simons term, Phys. Lett. B 366 (1996), 199 -204 | MR

[15] J. Han, Existence of topological multivortex solutions in the self-dual gauge theories, Proc. R. Soc. Edinb. A 130 (2000), 1293 -1309 | MR | Zbl

[16] J. Hong, Y. Kim, P.Y. Pac, Multi-vortex solutions of the Abelian Chern–Simons–Higgs theory, Phys. Rev. Lett. 64 (1990), 2230 -2233 | MR | Zbl

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

[18] K. Kimm, K. Lee, T. Lee, Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D 53 (1996), 4436 -4440

[19] K. Kimm, K. Lee, T. Lee, The self-dual Chern–Simons CP(N) models, Phys. Rev. Lett. B 380 (1996), 303 -307 | MR

[20] C.-S. Lin, S. Yan, Bubbling solutions for relativistic abelian Chern–Simons model on a torus, Commun. Math. Phys. 297 (2010), 733 -758 | MR | Zbl

[21] C.-S. Lin, S. Yan, Existence of bubbling solutions for Chern–Simons model on a torus, Arch. Ration. Mech. Anal. 207 (2013), 353 -392 | MR | Zbl

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

[23] B.J. Schroers, Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Lett. B 356 (1995), 291 -296 | MR

[24] J. Spruck, Y. Yang, The existence of nontopological solitons in the self-dual Chern–Simons theory, Commun. Math. Phys. 149 (1992), 361 -376 | MR | Zbl

[25] G. 'T Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979), 141 -160 | MR

[26] G. Tarantello, Uniqueness of selfdual periodic Chern–Simons vortices of topological-type, Calc. Var. Partial Differ. Equ. 29 (2007), 191 -217 | MR | Zbl

[27] G. Tarantello, Selfdual Gauge Field Vortices. An Analytical Approach, Prog. Nonlinear Differ. Equ. Appl. , Birkhäuser Boston, Inc., Boston (2008) | MR

[28] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monogr. Math. , Springer-Verlag, New York (2001) | MR

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