A New Method to prove the Existence, Non-existence, Multiplicity, Uniqueness, and Orbital Stability/Instability of standing waves for NLS with partial confinement

Abstract: We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction appears for some parameters. We also find different bifurcation phenomena from the cases with full confinement.