Condensate solutions of the self-dual O(3) Maxwell–Chern–Simons–Higgs equations with symmetric vacua

  • Jongmin Han
  • Kyungwoo SongEmail author


We consider a semilinear elliptic system, called the self-dual O(3) Maxwell–Chern–Simons–Higgs equations, on a two-dimensional flat torus arising from the O(3) sigma gauge field model. The system has three important parameters \(\tau \in [0,1]\), \(\kappa >0\) and \(q>0\). We focus on the case of \(\tau =1\) which yields a different structure from that of \(0 \le \tau <1\). Then, we show that the system possesses a solution for a sufficiently small \(\kappa \) and a very large q on a torus. In the proof, we develop a new method in order to use the topological degree theory. We also prove the Chern–Simons limit for our solutions, i.e., the convergence of solutions as \(q \rightarrow \infty \).

Mathematics Subject Classification

35J61 35Q75 81T13 



The authors thank the referee for helpful suggestions and comments. The research of Jongmin Han was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2018R1D1A1B07042681), and the research of Kyungwoo Song was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A1B03932448).


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Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsKyung Hee UniversitySeoulKorea

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