Two Normalized Solutions for the Chern–Simons–Schrödinger System with Exponential Critical Growth

Abstract

In this paper, we investigate normalized solutions for the Chern–Simons–Schrödinger system with a trapping potential $V(x)=\omega |x|^{2}$ and a exponential critical growth f(u). The solutions correspond to critical points of the underlying energy functional subject to the $L^{2}$-norm constraint, namely, $\int _{\mathbb {R}^{2}}|u|^{2}dx=c$ for $c>0$ given. Under some suitable assumptions on f,  we show that the system has at least two normalized solutions $u_{c},\hat{u}_{c}\in H^{1}(\mathbb {R}^{2})$, depending on the trapping frequency $\omega$ and the mass c, where $u_{c}$ is a ground state with positive energy and orbitally stable, while $\hat{u} _{c}$ is a high-energy solution with positive energy. In addition, the asymptotic behavior of the solution $u_{c}$ as $c\rightarrow 0$ is described.