On the vortices for the nonlinear Schrödinger equation in higher dimensions

We consider the nonlinear Schrödinger equation in n space dimensions [Formula: see text]and study the existence and stability of standing wave solutions of the form [Formula: see text]and [Formula: see text]For n=2k, (r ,θ ) are polar coordinates in [Formula: see text], j=1,2,…,k; for n=2k+1, (r ,θ ) are polar coordinates in [Formula: see text], (r ,θ ,z) are cylindrical coordinates in [Formula: see text], j=1,2,…,k-1. We show the existence of functions ϕ , which are constructed variationally as minimizers of appropriate constrained functionals. These waves are shown to be spectrally stable (with respect to perturbations of the same type), if 1<p<1+4/nThis article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.