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

We study the following nonlinear Schr\"odinger equation and we look for normalized solutions $(u,\mu)\in H^1(\R^N)\times\R$ for a given $m>0$ and $N\geq 2$ \[ -\Delta u + \mu u = g(u)\quad \hbox{in}\ \R^N, \qquad \frac{1}{2}\intRN u^2 dx = m. \] We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=\abs{s}^{p-1}s +h(s)$, $h(s)=o(|s|^p)$ as $s\sim 0$ and $s\sim\infty$. % The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s) = \abs{s}^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}\intRN\omega_1^2\, dx$ is the mass of a least energy solution $\omega_1$ of $-\Delta \omega+\omega=\omega^p$ in $\R^N$. 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 $s\sim\infty$. In contrast, we show non-existence results for $h(s)\not=o(s)$ ($s\sim 0$) under a suitable monotonicity condition.