Normalized ground states for the NLS equation with combined nonlinearities

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −Δu=λu+μ|u|q−2u+|u|p−2uin RN, N≥1, having prescribed mass ∫RN|u|2=a2. Under different assumptions on q0 and μ∈R we prove several existence and stability/instability results. In particular, we consider cases when [Formula presented] i.e. the two nonlinearities have different character with respect to the L2-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.