数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1743-1764.doi: 10.1007/s10473-022-0503-3

• 论文 • 上一篇    

A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM

Jin DENG, Benniao LI   

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, China
  • 收稿日期:2021-11-26 修回日期:2022-06-10 发布日期:2022-11-02
  • 通讯作者: Jin Deng,E-mail:jindeng_2016@126.com E-mail:jindeng_2016@126.com
  • 基金资助:
    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).

A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM

Jin DENG, Benniao LI   

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, China
  • Received:2021-11-26 Revised:2022-06-10 Published:2022-11-02
  • Contact: Jin Deng,E-mail:jindeng_2016@126.com E-mail:jindeng_2016@126.com
  • Supported by:
    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).

摘要: In this paper, we consider the Chern-Simons-Schrödinger system \begin{equation*}\left\{\begin{array}{lll} - \Delta u+\left[e^{2}|\mathbf{A}|^{2}+\left(V(x)+2e A_{0}\right)+2\left(1+\frac{\kappa q}{2 }\right) N\right] u+ q |u|^{p-2}u=0, \\ -\Delta N+\kappa^{2} q^{2} N+q\left(1+\frac{\kappa q}{2}\right) u^{2}=0, \\ \kappa\left(\partial_{1} A_{2}-\partial_{2} A_{1}\right)= - e u^{2}, \, \, \partial_{1} A_{1}+\partial_{2} A_{2}=0, \\ \kappa \partial_{1} A_{0}= e^{2} A_{2} u^{2}, \, \, \kappa \partial_{2} A_{0}= - e^{2} A_{1} u^{2}, \, \, \end{array} \right.{\rm (P)} \end{equation*} where $u \in H^{1}(\mathbb{R}^{2})$, $p \in (2, 4)$, $A_{\alpha}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ are the components of the gauge potential $(\alpha=0, 1, 2)$, $N: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a neutral scalar field, $V(x)$ is a potential function, the parameters $ \kappa, 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.

关键词: Chern-Simons-Schrödinger systems, ground state solution, variational method

Abstract: In this paper, we consider the Chern-Simons-Schrödinger system \begin{equation*}\left\{\begin{array}{lll} - \Delta u+\left[e^{2}|\mathbf{A}|^{2}+\left(V(x)+2e A_{0}\right)+2\left(1+\frac{\kappa q}{2 }\right) N\right] u+ q |u|^{p-2}u=0, \\ -\Delta N+\kappa^{2} q^{2} N+q\left(1+\frac{\kappa q}{2}\right) u^{2}=0, \\ \kappa\left(\partial_{1} A_{2}-\partial_{2} A_{1}\right)= - e u^{2}, \, \, \partial_{1} A_{1}+\partial_{2} A_{2}=0, \\ \kappa \partial_{1} A_{0}= e^{2} A_{2} u^{2}, \, \, \kappa \partial_{2} A_{0}= - e^{2} A_{1} u^{2}, \, \, \end{array} \right.{\rm (P)} \end{equation*} where $u \in H^{1}(\mathbb{R}^{2})$, $p \in (2, 4)$, $A_{\alpha}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ are the components of the gauge potential $(\alpha=0, 1, 2)$, $N: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a neutral scalar field, $V(x)$ is a potential function, the parameters $ \kappa, 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.

Key words: Chern-Simons-Schrödinger systems, ground state solution, variational method

中图分类号: 

  • 35J20