Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity

We study the following nonlinear Schrödinger equation and we look for normalized solutions $(\mu ,u)\in \mathbf{R}\times H^1({\mathbf{R}}^N)$ for a given $m>0$ and $N\ge 2$ $$-\mathrm{\Delta }u+\mu u=g(u)\text{in }{\mathbf{R}}^N,\frac{1}{2}{\int }_{{\mathbf{R}}^N}{u}^{2}dx=m\mathrm{.}$$ We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=\mathrm{\mid }s{\mathrm{\mid }}^{p-1}s+h(s)$, $h(s)=o(\mathrm{\mid }s{\mathrm{\mid }}^p)$ as $s\sim 0$ and $s\sim \mathrm{\infty }$. The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s)=\mathrm{\mid }s{\mathrm{\mid }}^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}{\int }_{{\mathbf{R}}^N}{\omega }_1^{2}dx$ is the mass of a least energy solution ${\omega }_1$ of $-\mathrm{\Delta }\omega +\omega ={\omega }^p$ in ${\mathbf{R}}^N$. We prove the existence of a positive solution for $m=m_1$ when $h$ has a sublinear growth at infinity, i.e., $h(s)=o(s)$ as $s\sim \mathrm{\infty }$. In contrast, we show non-existence results for $h(s)≠o(s)$ ($s\sim 0$) under a suitable monotonicity condition.