  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.
