Normalized solutions to the Chern-Simons-Schrödinger system

Abstract

In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schrödinger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, orbital stability of the set of minimizers to the minimization problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the minimization problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solutions. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, orbital instability of ground states is explored.