Bound states of the NLS equation on metric graphs with localized nonlinearities

TL;DR

This paper proves the existence of multiple bound states with prescribed mass for the nonlinear Schrödinger equation on metric graphs with localized nonlinearities, showing that the number of solutions increases with the mass.

Contribution

It demonstrates the existence of multiple solutions for the NLS on metric graphs with localized nonlinearities, a novel result in this setting.

Findings

At least k solutions exist for large enough mass

Solutions are constrained critical points of the energy functional

Energy estimates for solutions are provided

Abstract

We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schr\"odinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k, the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.