Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state.