Lack of ground state for NLS on bridge-type graphs

TL;DR

This paper proves that no ground state solutions exist for the nonlinear Schrödinger equation on bridge-type graphs with specific topologies and boundary conditions, highlighting limitations in the existence of energy minimizers.

Contribution

It establishes the nonexistence of ground states for NLS on a class of bridge-like graphs with Kirchhoff conditions, a novel result in the study of NLS on complex networks.

Findings

No ground states exist for NLS on bridge graphs with specified topology.

The proof applies to graphs with two halflines and four vertices at infinity.

Results impact understanding of energy minimization in quantum graphs.

Abstract

We prove the nonexistence of ground states for NLS on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass.