Existence and Stability of standing waves for supercritical NLS with a Partial Confinement

We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.Comment: Revised version, accepted on Comm. Math. Physic