CONCENTRATION PHENOMENON FOR THE L2 CRITICAL AND SUPER CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS IN ENERGY SPACES

We show that the blow up solution with data in energy spaces H1 of the nonlinear Schrödinger equation i ut + Δu + |u|αu = 0 has a concentration behavior in the critical Sobolev space Ḣsα (0 < sα = n/2 - 2/α ≤ 1) if the time variable tends to the blow up time. Moreover, the blow up points in ℝn are also quantified in the L2 critical case α = 4/n.