Existence of vortices in nonlinear optics

Optical propagation in nonlinear media and the formation of optical vortices as dark holes have been intensively studied in modern optical physics. In this paper, we prove the existence of different types of stationary vortex wave solutions of a general class for nonlinear Schrödinger equations. First, we prove the existence of positive radially symmetric solutions by solving a constrained minimization problem and give some lower estimate of the wave propagation constant. We then use a min-max technique to prove the existence of additional non-trivial solutions which arise as saddle-points of a corresponding indefinite action functional. At the request of the Editor-in-Chief and the authors this articles has been retracted. Due to an irreparable error in the arguments, the main results are not correct.