  We consider the nonlinear Schr\"odinger equation with combined
nonlinearities, where the leading term is an intracritical focusing power-type
nonlinearity, and the perturbation is given by a power-type defocusing one. We
completely answer the question wether the ground state energy, which is a
threshold between global existence and formation of singularities, is achieved.
For any prescribed mass, for mass-supercritical or mass-critical defocusing
perturbations, the ground state energy is achieved by a radially symmetric and
decreasing solution to the associated stationary equation. For mass-subcritical
perturbations, we show the existence of a critical prescribed mass, precisely
the mass of the unique, static, positive solution to the associated elliptic
equation, such that the ground state energy is achieved for any mass equal or
smaller than the critical one. Moreover, the ground state energy is not
achieved for mass larger than the critical one. As a byproduct of the
variational characterization of the ground state energy, we prove the existence
of blowing-up solutions in finite time, for any energy below the ground state
energy threshold.
