Ground state energy threshold and blow-up for NLS with competing nonlinearities

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