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

TL;DR

This paper investigates the existence of non-trivial bound states with prescribed mass for the supercritical nonlinear Schrödinger equation on compact metric graphs, using variational methods and blow-up analysis.

Contribution

It introduces a novel variational approach combining the monotonicity trick and second order min-max techniques for supercritical NLS on graphs.

Findings

Existence of bound states with prescribed mass established.

New variational framework applicable to supercritical regimes.

Analysis of blow-up behavior and Morse index of solutions.

Abstract

This paper is devoted to the existence of non-trivial bound states of prescribed mass for the mass-supercritical nonlinear Schr\"odinger equation on compact metric graphs. The investigation is based upon a general variational principle which combines the monotonicity trick and a min-max theorem with second order information, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.