Action versus energy ground states in nonlinear Schr\"odinger equations

TL;DR

This paper explores the relationship between energy and action ground states in nonlinear Schrödinger equations, revealing duality, conditions for equivalence, and properties of ground state levels.

Contribution

It establishes a duality between energy and action ground state levels via Legendre-Fenchel transform and analyzes the differentiability and convexity properties of the action level.

Findings

Energy ground state level is the Legendre-Fenchel transform of the action ground state level.

All action ground states with a given frequency share the same mass and are energy ground states if an energy ground state exists.

Action ground state level may be non-convex and its differentiability involves the mass of ground states.

Abstract

We investigate the relations between normalized critical points of the nonlinear Schr\"odinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold. Our first general result is that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre-Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We prove that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers.