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A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM
Jin DENG, Benniao LI
Acta mathematica scientia,Series B. 2022, 42 (5):
1743-1764.
DOI: 10.1007/s10473-022-0503-3
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.
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