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We study the nature of the nonlinear Schrödinger equation ground states on the product spaces [math] , where [math] is a compact Riemannian manifold. We prove that for small [math] masses the ground states coincide with the corresponding [math] ground states. We also prove that above a critical mass the ground states have nontrivial [math] dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.

International audience; We study the long-time behavior of the 3-dimensional repulsive nonlinear Hartree equation with an external attractive Coulomb potential $-Z/|x|$, which is a nonlinear model for the quantum dynamics of an atom. We show that, after a sufficiently long time, the average number of electrons in any finite ball is always smaller than 4Z (respectively 2Z in the radial case). This is a time-dependent generalization of a celebrated result by E.H. Lieb on the maximum negative ionization of atoms in the stationary case. Our proof involves a novel positive commutator argument (based on the cubic weight $|x|^3$) and our findings are reminiscent of the RAGE theorem. In addition, we prove a similar universal bound on the local kinetic energy. In particular, our main result means that, in a weak sense, any solution is attracted to a bounded set in the energy space, whatever the size of the initial datum. Moreover, we extend our main result to Hartree--Fock theory and to the linear many-body Schrödinger equation for atoms.