# A Ground State Solution to the Chern-Simons-Schrödinger System

## Abstract

In this paper, we consider the Chern-Simons-Schrödinger system

$$\left\{{\matrix{{- \Delta u + \left[{{e^2}{{\left| {\bf{A}} \right|}^2} + \left({V\left(x \right) + 2e{A_0}} \right) + 2\left({1 + {{\kappa q} \over 2}} \right)N} \right]u + q{{\left| u \right|}^{p - 2}}u = 0,} \hfill \cr {- \Delta N + {\kappa ^2}{q^2}N + q\left({1 + {{\kappa q} \over 2}} \right)\,\,{u^2} = 0,} \hfill \cr {\kappa \left({{\partial _1}{A_2} - {\partial _2}{A_1}} \right) = - e{u^2},\,\,\,\,{\partial _1}{A_1} + {\partial _2}{A_2} = 0,} \hfill \cr {\kappa {\partial _1}{A_0} = {e^2}{A_2}{u^2},\,\,\,\kappa {\partial _2}{A_0} = - {e^2}{A_1}{u^2},} \hfill \cr}} \right.$$

where uH1(ℝ2), p ∈ (2, 4), Aα: ℝ2 → ℝ are the components of the gauge potential (α = 0, 1, 2), N: ℝ2 → ℝ is a neutral scalar field, V(x) is a potential function, the parameters κ, q > 0 represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and e > 0 is the coupling constant. In this paper, the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem. The ground state solution of the problem (P) is obtained by using the variational method.

This is a preview of subscription content, access via your institution.

## References

1. Berge L, de Bouard A, Saut J C. Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation. Nonlinearity, 1995, 8: 235–253

2. Byeon J, Huh H. On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations. J Differential Equations, 2016, 261: 1285–1316

3. Byeon J, Huh H, Seok J. Standing waves of nonlinear Schrödinger equations with the gauge field. J Funct Anal, 2012, 263: 1575–1608

4. Cunha P L, d’Avenia P, Pomponio A, Siciliano G. A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity. Nonlinear Differ Equ Appl, 2015, 22: 1831–1850

5. Chen S T, Zhang B Z, Tang X H. Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons- Schrödinger system in H1(ℝ2). Nonlinear Analysis, 2019, 185: 68–96

6. d’Avenia P, Pomponio A. Standing waves for a Schrödinger-Chern-Simons-Higgs system. J Differential Equations, 2020, 268: 2151–2162

7. Deng J. The existence of solutions for the Schrödinger Chern-Simons-Higgs system. Acta Mathematica Scientia, 2021, 41A(6): 1768–1778

8. Deng J, Li B N, Yang J F. Solutions to strongly indefinite Chern-Simons-Schrödinger system. Preprint

9. Dunne G V, Trugenberger C A. Self-duality and nonrelativistic Maxwell-Chern-Simons solitons. Phys Rev D, 1991, 43: 1323–1331

10. Huh H. Standing waves of the Schrödinger equation coupled with the Chern-Simons gauged field. J Math Phys, 2012, 53: 063702

11. Han J, Huh H, Seok J. Chern-Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field. J Funct Anal, 2014, 266: 318–342

12. Han J, Song K. On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains. J Math Anal Appl, 2009, 350: 1–8

13. Jackiw R, Pi S. Classical and quantal nonrelativistic Chern-Simons theory. Phys Rev D, 1990, 42: 3500–3513

14. Jackiw R, Pi S. Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys Rev Lett, 1990, 64: 2969–2972

15. Jackiw R, Pi S. Self-dual Chern-Simons solitons. Progr Theoret Phys Suppl, 1992, 107: 1–40

16. Jackiw R, Templeton S. How super-renormalizable interactions cure their infrared divergences. Phys Rev D, 1981, 23: 2291–304

17. Li L Y, Yang J F, Yang J G. Solutions to Chern-Simons-Schrödinger systems with external potential. Dis Conti Dyn Sys S, 2021, 14: 1967–1981

18. Martina L, Pashaev O K, Soliani G. Chern-Simons gauge field theory of two dimensional ferromagnets. Phys Rev B, 1993, 48: 15787–15791

19. Pomponio A, Ruiz D. A variational analysis of a gauged nonlinear Schrödinger equation. J Eur Math Soc, 2015, 17: 1463–1486

20. Pomponio A, Ruiz D. Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc Var Partial Differential Equations, 2015, 53: 289–316

21. Tan J, Li Y, Tang C. The existence and concentration of ground state solutions for Chern-Simons-Schrödinger systems with a steep well potential. Acta Mathematica Scientia, 2022, 42B(3): 1125–1140

22. Wan Y, Tan J. Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition. J Math Anal Appl, 2014, 415: 422–434

23. Wan Y, Tan J. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete Contin Dyn Syst, 2017, 37: 2765–2786

24. Willem M. Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, 24. Boston MA: Birkhäuser Boston, Inc, 1996

## Acknowledgements

The authors would like to thank Professor Jianfu Yang for many helpful discussions and comments.

## Author information

Authors

### Corresponding author

Correspondence to Jin Deng.