Sign-changing prescribed mass solutions for $L^2$-supercritical NLS on compact metric graphs

TL;DR

This paper proves the existence of multiple sign-changing solutions with prescribed mass for a supercritical nonlinear Schrödinger equation on compact metric graphs, introducing new methods and identifying bifurcation points.

Contribution

It provides the first multiplicity results for prescribed mass solutions in the supercritical regime on compact metric graphs, using novel linking and gradient flow techniques.

Findings

Multiple sign-changing solutions exist in the supercritical mass regime.

Any eigenvalue of the linear operator is a bifurcation point.

The methods can be adapted to other bounded domain problems.

Abstract

This paper is devoted to the existence of multiple sign-changing solutions of prescribed mass for a mass-supercritical nonlinear Schr\"odinger equation set on a compact metric graph. In particular, we obtain, in the supercritical mass regime, the first multiplicity result for prescribed mass solutions on compact metric graphs. As a byproduct, we prove that any eigenvalue of the associated linear operator is a bifurcation point. Our approach relies on the introduction a new kind of link and on the use of gradient flow techniques on a constraint. It can be transposed to other problems posed on a bounded domain.