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

In this paper, we are interested in standing waves with a vortex for the nonlinear Chem-Simons-Schrodinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant lambda > 0, representing the strength of the interaction potential, varies. We prove every standing wave is trivial if lambda is an element of (0, 1), every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS lambda = 1 and for every positive integer N, there is a nontrivial standing wave with a vortex point of order N if lambda > 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. (C) 2016 Elsevier Inc. All rights reserved.