Multiple positive bound states for NLS equations on noncompact metric graphs with an attractive potential

TL;DR

This paper proves the existence of multiple bound states for the nonlinear Schrödinger equation on noncompact metric graphs with an attractive potential, especially when the mass is large, highlighting differences from ground state behavior.

Contribution

It demonstrates the existence of multiple positive bound states on metric graphs with attractive potentials, depending on the graph's structure and potential assumptions.

Findings

Number of bound states equals the number of bounded edges for large mass

Bound states exist under certain potential conditions

Ground state may not exist for all masses

Abstract

In this paper, we establish the existence of bounded states and geometrically distinct solutions for the subcritical NLS equation with attractive potential on metric graphs $\mathcal{G}$ when the mass $\mu$ is large enough.We show that the NLS equation exists at least as many bound states of mass $\mu$ as the number of bounded edges of $\mathcal{G}$ if the attractive potential satisfies some suitable assumptions.It is worth noting that this is different from the case of ground state, which on some graphs may fail to exist for every value of $\mu$.