The nonlinear Schrödinger equation ground states on product spaces

Abstract

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces ${ℝ}^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding ${ℝ}^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.