Non-uniqueness of normalized ground states for nonlinear Schr\"odinger equations on metric graphs

TL;DR

This paper demonstrates that on certain metric graphs, normalized ground states for nonlinear Schrödinger equations are not unique near the critical nonlinearity power, revealing complex solution structures.

Contribution

It establishes non-uniqueness of normalized ground states for nonlinear Schrödinger equations on metric graphs near the critical exponent, a novel insight into their solution landscape.

Findings

Non-uniqueness of normalized ground states near critical nonlinearity.

Existence of action ground states not corresponding to normalized ground states.

Non-uniqueness occurs at specific mass values for graphs with ground states at every mass.

Abstract

We establish general non-uniqueness results for normalized ground states of nonlinear Schr\"odinger equations with power nonlinearity on metric graphs. Basically, we show that, whenever in the $L^2$-subcritical regime a graph hosts ground states at every mass, for nonlinearity powers close to the $L^2$-critical exponent $p=6$ there is at least one value of the mass for which ground states are non-unique. As a consequence, we also show that, for all such graphs and nonlinearities, there exist action ground states that are not normalized ground states.