Normalized ground states for the NLS equation with combined nonlinearities

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities $$-\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,}$$ having prescribed mass $$\int_{\mathbb{R}^N} |u|^2 = a^2.$$ Under different assumptions on $q<p$, $a>0$ and $\mu \in \mathbb{R}$ we prove several existence and stability/instability results. In particular, we consider cases when $$2<q \le 2+ \frac{4}{N} \le p<2^*, \quad q \neq p,$$ i.e. the two nonlinearities have different character with respect to the $L^2$-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.