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

TL;DR

This paper introduces a novel method to analyze the existence, multiplicity, and stability of standing waves in nonlinear Schrödinger equations with partial confinement, addressing open problems and revealing new bifurcation phenomena.

Contribution

The authors develop a new approach that removes the subcritical hypothesis and provides comprehensive results on existence, non-existence, and stability, including solutions' uniqueness and bifurcation behavior.

Findings

Confirmed the existence of multiple solutions beyond previous conjectures.

Identified conditions for stability and instability of standing waves.

Discovered dimension reduction and new bifurcation phenomena under 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.