On standing waves with a vortex point of order N for the nonlinear Chern–Simons–Schrödinger equations

In this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrödinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant λ>0λ>0, representing the strength of the interaction potential, varies. We prove every standing wave is trivial if λ∈(0,1)λ∈(0,1), every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS if λ=1λ=1 and for every positive integer N, there is a nontrivial standing wave with a vortex point of order N   if λ>1λ>1. We also provide some classes of interaction potentials under which the nonexistence of standing waves and the existence of a standing wave with a vortex point of order N are proved.