Existence of vortices for Schrödinger equations with logarithmic and saturable nonlinearity

Abstract

In this paper, we study the existence of stationary vortex wave solutions of two kinds of nonlinear Schrödinger equations. For the first one, which is equipped with logarithmic nonlinearity arising from Bose–Einstein condensation, we consider two types of boundary value problems. In both cases, we establish the existence of positive solutions through a direct minimization method. For the second one, with a saturable nonlinearity originating from geometric optics, we use a constrained minimization approach to establish the existence of vortex wave solutions. Moreover, some explicit estimates for the bound of the wave propagation constant are derived.