Normalized Solutions to Schr\"{o}dinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of positive solutions with prescribed mass for nonlinear Schrödinger equations in bounded domains using a global bifurcation approach, covering various nonlinearities and autonomous or non-autonomous cases.

Contribution

It generalizes previous results by handling a wider class of nonlinearities and domain types, providing new existence and stability results for solutions.

Findings

Established existence, nonexistence, and multiplicity of solutions.

Proved orbital stability and instability results.

Extended bifurcation analysis to non-autonomous and mixed nonlinearities.

Abstract

We obtain the existence, nonexistence and multiplicity of positive solutions with prescribed mass for nonlinear Schr\"{o}dinger equations in bounded domains via a global bifurcation approach. The nonlinearities in this paper can be mass supercritical, critical, subcritical or some mixes of these cases, and the equation can be autonomous or non-autonomous. This generalizes a result in Noris, Tavares and Verzini [\emph{Anal. PDE}, 7 (8) (2014) 1807-1838], where the equation is autonomous with homogeneous nonlinearities. Besides, we have proven some orbital stability or instability results.