Optical vortices arise as phase singularities of the light fields and are ofcentral interest in modern optical physics. In this paper, some existencetheorems are established for stationary vortex wave solutions of a generalclass of nonlinear Schr\"{o}dinger equations. There are two types of results.The first type concerns the existence of positive-radial-profile solutionswhich are obtained through a constrained minimization approach. The second typeaddresses the existence of saddle-point solutions through amountain-pass-theorem or min-max method so that the wave propagation constantmay be arbitrarily prescribed in an open interval. Furthermore some explicitestimates for the lower bound and sign of the wave propagation constant withrespect to the light beam power and vortex winding number are also derived forthe first type solutions.
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