Ground states for coupled NLS equations with double power nonlinearities

We study the existence of ground states (minimizers of an energy under a fixed masses constraint) for a system of coupled NLS equations with double power nonlinearities (classical and point). We prove that the presence of at least one subcritical power parameter guarantees existence of a ground state for masses below specific values. Moreover, we show that each ground state is given by a pair of strictly positive functions (up to rotation). Using the concentration-compactness approach, under certain restrictions we prove the compactness of each minimizing sequence. One of the principal peculiarities of the model is that in presence of critical power parameters existence of ground states depends on concrete mass parameters related with the optimal function for the Gagliardo–Nirenberg inequality.