Normalized ground states for NLS equations with mass critical nonlinearities

We study normalized solutions $(\mu ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to nonlinear Schrödinger equations $$-\mathrm{\Delta }u+\mu u=g(u)\text{in}{\mathbb{R}}^N,\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^{2}dx=m,$$ 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 $g(s)\sim \mathrm{\mid }s{\mathrm{\mid }}^{p-1}s$ as $s\sim 0$ and $s\sim \mathrm{\infty }$, 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 $\underset{‾}{b}\le 0\le \bar{b}$ of the Lagrangian functional. In this paper we furnish explicit examples of $g$ satisfying $\underset{‾}{b}<0<\bar{b}$, $\underset{‾}{b}=0<\bar{b}$ and $\underset{‾}{b}<0=\bar{b}$; notice that $\underset{‾}{b}=0=\bar{b}$ in the power case $g(t)=\mathrm{\mid }t{\mathrm{\mid }}^{p-1}t$. 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 $g$.