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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.

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Acknowledgements

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

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Correspondence to Jin Deng.

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Jin Deng was partially supported by NSFC (12161044) and Natural Science Foundation of Jiangxi Province (20212BAB211013), Benniao Li was partially supported by NSFC (12101274) and Doctoral Research Startup Foundation of Jiangxi Normal University (12020927).

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Deng, J., Li, B. A Ground State Solution to the Chern-Simons-Schrödinger System. Acta Math Sci 42, 1743–1764 (2022). https://doi.org/10.1007/s10473-022-0503-3

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Key words

  • Chern-Simons-Schrödinger systems
  • ground state solution
  • variational method

2010 MR Subject Classification

  • 35J20
  • 35Q40
  • 35Q51