Two Normalized Solutions for the Chern-Simons-Schrodinger System with Exponential Critical Growth

In this paper, we investigate normalized solutions for the Chern-Simons-Schrodinger 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, integral(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), u (c)is an element of H-1(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 u (c) is a high-energy solution with positive energy. In addition, the asymptotic behavior of the solution u(c) as c -> 0 is described.