A global branch approach to normalized solutions for the Schrödinger equation

Abstract

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schrödinger equation of the form \begin{equation*} -Δu+λu=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 $\lambda \to 0^{+}$ or $\lambda \to +\infty$ and the existence of an unbounded continuum of solutions in $(0,+\infty )\times H^1({\mathbb{R}}^N)$.

This version is the final one, corresponding to the paper now published in Journal de Mathématiques Pures et Appliquées