A global branch approach to normalized solutions for the Schr\"odinger equation

TL;DR

This paper investigates the existence, non-existence, and multiplicity of positive solutions with prescribed mass for a Schrödinger equation, using a global branch approach that unifies different nonlinear regimes.

Contribution

It introduces a unified method to analyze solutions across mass subcritical, critical, and supercritical cases, including asymptotic behaviors and an unbounded continuum of solutions.

Findings

Existence of solutions for various nonlinearities.

Asymptotic analysis as λ approaches 0 or infinity.

Presence of an unbounded continuum of solutions.

Abstract

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schr\"odinger equation of the form \begin{equation*} -\Delta u+\lambda u=g(u), \quad u \in H^1(\mathbb{R}^N), \, N \geq 1. \end{equation*} Our approach permits to handle in a unified way nonlinearities $g(s)$ which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as $\lambda\rightarrow 0^+$ or $\lambda\rightarrow +\infty$ and the existence of an unbounded continuum of solutions in $(0, + \infty) \times H^1(\mathbb{R}^N)$.