In this talk I will discuss a generalization of the usual nonlinear Schrdinger equation to systems of $N$ orthonormal functions. We can prove the existence of ground states for infinitely many $N$'s (including $N=2$) when the exponent $p$ of the nonlinearity is less than $\min(2,1+2/d)$, in dimension $d\geq1$. On the contrary, in dimension $d=1$ we show that there is no minimizer for all $N\geq2$ when $p=2$. Links with best constants in the Lieb-Thirring inequality will also be mentioned. Based on joint works with Rupert L. Frank, David Gontier & Faizan Q. Nazar.