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The analysis of solutions for Maxwell–Chern–Simons O(3) sigma model

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Abstract

In this paper, we prove the uniqueness of topological solutions for the self-dual Maxwell–Chern–Simons O(3) sigma model under the conditions on Chern–Simons coupling parameter and the charge of electron. Besides, we also provide the quantitative analysis of radial solutions of all types for single vortex-point case.

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References

  1. Bogomol’nyi, A.-A.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976)

    MathSciNet  Google Scholar 

  2. Chen, Z.-Y.: Topological solutions for self-dual Chern-Simons \(CP(1)\) model with large Chern-Simons coupling constant. Proc. Am. Math. Soc. 144, 191–203 (2016)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z.-Y.: On the Asymptotic Behavior of Solutions for the Self-Dual Maxwell–Chern–Simons \(O(3)\) Sigma Model. (Submitted)

  4. Chen, Z.-Y., Chern, J.-L.: Sharp Range of Flux and the Structure of Solutions for Self-Dual Chern–Simons \(CP(1)\) Model. (Submitted)

  5. Chen, Z.-Y., Chern, J.-L.: Uniqueness of topological multivortex solutions in the Maxwell-Chern-Simons Model. J. Funct. Anal. 270(6), 2073–2125 (2016)

    Article  MathSciNet  Google Scholar 

  6. 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. 296, 323–351 (2010)

    Article  MathSciNet  Google Scholar 

  7. Chen, Z.-Y., Chern, J.-L., Shi, J., Tang, Y.-L.: On the uniqueness and structure of solutions to a coupled elliptic system. J. Differ. Equ. 249, 3419–3442 (2010)

    Article  MathSciNet  Google Scholar 

  8. Chern, J.-L., Chen, Z.-Y., Tang, Y.-L., Lin, C.-S.: Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system. J. Differ. Equ. 246, 3704–3714 (2009)

    Article  MathSciNet  Google Scholar 

  9. Chan, H., Fu, C.-C., Lin, C.-S.: Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation. Commun. Math. Phys. 231, 189–221 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Chen, C.-C., Lin, C.-S.: Uniqueness of the ground state solutions of \(\Delta u+f(u)=0\) in \({{ R}}^n, n\ge 3\). Commun. Partial Differ. Equ. 16, 1549–1572 (1991)

    Article  Google Scholar 

  12. Chae, D., Nam, H.-S.: On the condensate multivortex solutions of the self-dual Maxwell-Chern-Simons CP(1) model. Anna. Henri Poincaré Anal. 2, 887–906 (2001)

    Article  Google Scholar 

  13. Choe, K., Nam, H.-S.: Existence and uniqueness of topological multivortex solutions in the self-dual Chern-Simons CP(1) model. Nonlinear Anal. 66, 2794–2813 (2007)

    Article  MathSciNet  Google Scholar 

  14. Chiacchio, F., Ricciardi, T.: Multivortex vortices for a self-dual Maxwell-Chern-Simons CP(1) model. NoDEA. 13, 563–584 (2007)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Han, J., Lin, C.-S.: Multiplicity for a self-dual condensate solutions in the Maxwell-Chern-Simons gauged O(3) sigma model. Commun. PDE. 39, 1424–1450 (2014)

    Article  Google Scholar 

  17. Han, J., Nam, H.-S.: On the topological multivortex solutions of the self-dual Maxwell-Chern-Simons gauged O(3) sigma model. Lett. Math. Phys. 73, 17–31 (2005)

    Article  MathSciNet  Google Scholar 

  18. Han, J., Song, K.: Existence and asymptotic of topological solutions in the self-dual Maxwell-Chern-Simons \(O(3)\) sigma model. J. Differ. Equ. 250, 204–222 (2011)

    Article  MathSciNet  Google Scholar 

  19. Kwong, M.-K.: Uniqueness of positive solutions of \(\Delta u-u+u^p=0\) in \({\mathbf{R}}^n\). Arch. Ration. Mech. Anal. 105, 243–266 (1989)

    Article  Google Scholar 

  20. Kim, K., Lee, K., Lee, T.: Anyonic Bogomol’nyi solitons in a gauged \(O(3)\) sigma model. Phys. Rev. D 53, 4436–4440 (1996)

    Article  Google Scholar 

  21. Mcleod, K.: Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \({\mathbf{R}}^n\). II. Trans. AMS. 339(2), 495–505 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Ricciardi, T.: Multiplicity for a nonlinear fourth-order elliptic equation in Maxwell-Chern-Simons vortex theory. Differ. Integral Equ. 17, 369–390 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Schroers, B.-J.: Bogomol’nyi solitons in a gauged \(O(3)\) sigma model. Phys. Lett. B 356, 291–296 (1995)

    Article  MathSciNet  Google Scholar 

  24. Yanagida, E.: Structure of radial solutions to \(\Delta u+K(|x|)|u|^{p-1}u=0\) in \({{ R}}^n\). SIAM. J. Math. Anal. 27, 997–1014 (1996)

    Article  MathSciNet  Google Scholar 

  25. Yang, S.-G., Chen, Z.-Y., Chern, J.-L.: The solution structure of the \(O(3)\) sigma model in a Maxwell-Chern-Simons theory. J. Math. Phys. 58, 071503 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees for helpful suggestions and comments to improve the exposition of the article. The work is partly supported by the Ministry of Science and Technology of Taiwan under the grant MOST 106-2628-M-018-001-MY4, MOST 107-2115-M-008-005-MY3 and National Center for Theoretical Science (NCTS).

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Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, Changhua, 500, Taiwan

    Zhi-You Chen

  2. Department of Mathematics, National Central University, Chung-Li, 32001, Taiwan

    Jann-Long Chern

Authors
  1. Zhi-You Chen
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  2. Jann-Long Chern
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Correspondence to Jann-Long Chern.

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Communicated by Y. Giga.

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Z.-Y. Chen: work partially supported by Ministry of Science and Technology of Taiwan under Grant No. MOST 106-2628-M-018-001-MY4.

J.-L. Chern: work partially supported by Ministry of Science and Technology of Taiwan under Grant No. MOST 107-2115-M-008-005-MY3.

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Chen, ZY., Chern, JL. The analysis of solutions for Maxwell–Chern–Simons O(3) sigma model. Calc. Var. 58, 147 (2019). https://doi.org/10.1007/s00526-019-1590-4

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