We consider the nonlinear Schrödinger equation in n space dimensions iut+Δu+|u|p−1u=0,x∈Rn,t>0 and study the existence and stability of standing wave solutions of the form {eiwtei∑kj=1mjθjϕw(r1,r2,…,rk),eiwtei∑kj=1mjθjϕw(r1,r2,…,rk,z),n=2kn=2k+1 for n=2k, (rj,θj) are polar coordinates in R2, j=1,2,…,k; for n=2k+1, (rj,θj) are polar coordinates in R2, (rk,θk,z) are cylindrical coordinates in R3, j=1,2,…,k−1. We show the existence of such solutions as minimizers of a constrained functional and conclude from there that such standing waves are stable if 1<p<1+4/n.