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. 
Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa
math.AP math-ph math.MP