Normalized ground states for NLS equations with mass critical nonlinearities

TL;DR

This paper investigates normalized solutions to mass-critical nonlinear Schrödinger equations, analyzing the existence, non-existence, and properties of solutions with specific nonlinearities and their associated energy levels.

Contribution

It provides explicit examples of nonlinearities with critical growth, characterizes minimax values, and explores conditions for positive solutions and minimal energy solutions.

Findings

Explicit examples of nonlinearities with different minimax value configurations.

Conditions under which solutions with minimal energy exist or do not exist.

Analysis of perturbations of the nonlinearity for positive solution existence.

Abstract

We study normalized solutions $(\mu,u)\in \mathbb{R} \times H^1(\mathbb{R}^N)$ to nonlinear Schr\"odinger equations $$ -\Delta u + \mu u = g(u)\quad \hbox{in}\ \mathbb{R}^N, \qquad \frac{1}{2}\int_{\mathbb{R}^N} u^2 dx = m, $$ where $N\geq 2$ and the mass $m>0$ is given. Here $g$ has an $L^2$-critical growth, both at the origin and at infinity, that is $g(s)\sim |s|^{p-1}s$ as $s\sim 0$ and $s\sim\infty$, where $p=1+\frac{4}{N}$. We continue the analysis started in [Cingolani-Gallo-Ikoma-Tanaka, 2024], where we found two (possibly distinct) minimax values $\underline{b} \leq 0 \leq \overline{b}$ of the Lagrangian functional. In this paper we furnish explicit examples of $g$ satisfying $\underline{b}<0<\overline{b}$, $\underline{b}=0<\overline{b}$ and $\underline{b}<0=\overline{b}$; notice that $\underline{b}=0=\overline{b}$ in the power case $g(t)=|t|^{p-1}t$. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on $g$ to obtain the existence of a positive solution for perturbations of $g$.