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

In this paper, we investigate normalized solutions for the Chern–Simons–Schrödinger system with a trapping potential V(x)=ω |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, ∫ _ℝ^2|u|^2dx=c for c>0 given. Under some suitable assumptions on f , we show that the system has at least two normalized solutions u_c,û_c∈ H^1(ℝ^2) , depending on the trapping frequency ω and the mass c , where u_c is a ground state with positive energy and orbitally stable, while û _c is a high-energy solution with positive energy. In addition, the asymptotic behavior of the solution u_c as c→ 0 is described.