Normalized ground states for NLS equations with mass critical nonlinearities

Abstract

We study normalized solutions $(\hat{I}¼,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to nonlinear Schrödinger equations $$-\hat{I}”u+\hat{I}¼u=g(u)in{\mathbb{R}}^N,\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^{2}dx=m,$$ where $N\ge 2$ 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 $\underline{b}\le 0\le \overline{b}$ 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 $g$.