Existence and Concentration of Semi-classical Ground State Solutions for Chern–Simons–Schrödinger System

Abstract

In this paper, we consider the equation $$\begin{aligned} -\varepsilon ^{2}\Delta u+ V(x)u+\left( A_{0}(u)+A_{1}^{2}(u)+A_{2}^{2}(u)\right) u=f(u) \ \ \ \ \mathrm {in} ~ H^{1}({\mathbb {R}}^{2}), \end{aligned}$$ 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 $$\varepsilon \rightarrow 0$$ , the global maximum point $$x_\varepsilon $$ of $$u_\varepsilon $$ must concentrate at the global minimum point $$x_0$$ of V.