Solutions to Sobolev Supercritical Nonlinear Schrodinger Equations on an Annulus via a Hopf Reduction Method

TL;DR

This paper develops a Hopf fibration-based reduction method to establish existence and multiplicity of positive solutions for Sobolev supercritical nonlinear Schrödinger equations on an annulus, revealing new mass thresholds.

Contribution

It introduces a novel reduction technique transforming the problem into a lower-dimensional setting, enabling analysis of solution existence in supercritical regimes.

Findings

Existence of a global minimizer in mass subcritical/critical regimes.

Presence of two solutions in supercritical regimes: a local minimizer and a mountain pass solution.

Identification of a new mass critical threshold for solutions.

Abstract

This paper investigates the existence of positive solutions with a prescribed mass for nonlinear Schrodinger equations on an annulus, possibly in the Sobolev supercritical regime. A reduction method based on the Hopf fibration is used to transform the problem into a lower-dimensional one. We obtain a new mass critical threshold and we show that in the new mass subcritical or critical regimes there exists a positive solution which corresponds to a global minimizer, while in the mass supercritical regime, there exists two positive solutions which correspond to a local minimizer and a mountain pass solution respectively. Some other problems are also discussed in this paper.