Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs

TL;DR

This paper proves the existence of positive normalized solutions for $L^2$-supercritical nonlinear Schrödinger equations on certain noncompact metric graphs, highlighting the influence of graph topology and mass size.

Contribution

It extends the analysis of normalized solutions to $L^2$-supercritical regimes on noncompact graphs, using mountain pass techniques and addressing compactness issues.

Findings

Existence of solutions for small mass on periodic and certain noncompact graphs.

Solutions with small mass have positive energy, aiding compactness.

Large mass solutions may not have positive energy, indicating different behavior.

Abstract

We consider the existence of normalized solutions to nonlinear Schr\"odinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive solutions on two classes of graphs: periodic graphs, and noncompact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an $L^2$ supercritical nonlinearity extended on the whole graph and unraveling the role of topology in the existence of solutions.