Existence and concentration of semi-classical ground state solutions for Chern-Simons-Schr\"{o}dinger system

• 1、School of Mathematics and Statistics, Southwest University, Chongqing 400715

Abstract：In this paper, we study the equation\begin{equation*} -\varepsilon^{2}\Delta u+ V(x)u+\left(A_{0}(u)+A_{1}^{2}(u)+A_{2}^{2}(u)\right)u=f(u) \ \ \ \ \mathrm{in} ~ H^{1}(\mathbb{R}^{2}),\end{equation*}where $\varepsilon$ is a small parameter, $V$ is the external potential,$A_i(i=0,1,2)$ is the gauge field and $f\in C(\mathbb{R}, \mathbb{R})$ is 5-superlinear growth.By using variational methods and analytic technique, we prove that this system possesses a ground state solution $u_\varepsilon$.Moreover, our results show that, as $\varepsilon\to 0$, the global maximum point $x_\varepsilon$ of $u_\varepsilon$ must concentrate at the global minimum point $x_0$ of $V$.

keywords： Chern-Simons-Schr\"{o}dinger system Semi-classical solution Ground state solutions Concentration Variational methods

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