An action approach to nodal and least energy normalized solutions for nonlinear Schr\"odinger equations

TL;DR

This paper introduces a novel approach to studying normalized solutions of nonlinear Schrödinger equations by analyzing the masses of ground states, leading to new existence results for nodal and least energy solutions across different regimes.

Contribution

It provides a complete characterization of ground state masses and links normalized solutions to action ground states, advancing understanding of solution structures in nonlinear Schrödinger equations.

Findings

Existence of normalized nodal solutions for all subcritical masses.

Existence of solutions for a range of critical and supercritical masses.

Identification of conditions under which normalized solutions are action ground states.

Abstract

We develop a new approach to the investigation of normalized solutions for nonlinear Schr\"odinger equations based on the analysis of the masses of ground states of the corresponding action functional. Our first result is a complete characterization of the masses of action ground states, obtained via a Darboux-type property for the derivative of the action ground state level. We then exploit this result to tackle normalized solutions with a twofold perspective. First, we prove existence of normalized nodal solutions for every mass in the $L^2$-subcritical regime, and for a whole interval of masses in the $L^2$-critical and supercritical cases. Then, we show when least energy normalized solutions/least energy normalized nodal solutions are action ground states/nodal action ground states.