Existence Of Vortices For Nonlinear Schr(?)dinger Equations

In this paper,we study the existence of vortices for three kinds of nonlinear Schr(?)dinger equations.For a class of Gross-Pitaevskii equation,we consider the existence of vortices on infinite intervals.Firstly,according to the symmetric mountain-path theorem,the equation has infinite pairs solutions including positive solutions under homogeneous boundary conditions.Secondly,we obtain the decay estimates of solutions at infinity through analytical techniques.Finally,the existence of vortices under non-homogeneous boundary conditions is proved by variational method.Next,we concentrate on the existence of vortices for two geometric optical models.Some conclusions about the existence of vortices in model ? are siven.Then,the con-clusion is extended to the case where the nonlinear term is a polynomial with 4m+1(m ? Z+)degree.The vortices in this case are proved by variational method.For model ?,we establish the existence theorem of positive-radial-symmetry solutions on bounded intervals through the constrained variational method.Furthermore some explicit estimates for the upper and lower bounds of wave propagation constants ? are derived.