On the notion of ground state for nonlinear Schr\"odinger equations on metric graphs

TL;DR

This paper investigates the existence and properties of ground states and least action solutions for nonlinear Schr"odinger equations on metric graphs, revealing that all possible cases of their existence and levels can occur.

Contribution

It provides a comprehensive analysis of the different scenarios for ground states and least action solutions on metric graphs, including new multiplicity results.

Findings

All four cases of existence and levels of solutions occur

Established new multiplicity results for positive solutions

Analyzed doubly constrained variational problems in depth

Abstract

We compare ground states for the nonlinear Schr\"odinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.