| Abstract: |
| We study the existence of normalized solutions for nonlinear Schr\"odinger equations:
$$ -\Delta u + \mu u = g(u) \quad \hbox{in}\ {\mathbf R}^N, \qquad \int_{{\mathbf R}^N} |u|^2\, dx=m,
$$
where $N\geq 2$, $g\in C({\mathbf R})$, $m>0$ are given and $\mu>0$, $u\in H^1({\mathbf R}^N)$ are unknown.
We consider the situation
$$ g(s) \sim |s|^{4/N}s \quad \hbox{as} \ s \sim 0 \ \hbox{and}\ s\sim \pm \infty
$$
and we show the existence of positive normalized solutions for a suitable $m>0$.
This is a joint work with Silvia Cingolani, Marco Gallo and Norihisa Ikoma. |
|