L2 CONCENTRATION OF BLOW-UP SOLUTIONS FOR THE NONLINEAR SCHRODINGER-EQUATION WITH CRITICAL POWER NONLINEARITY

We consider the solution of the nonlinear Schrödinger equation i ∂u ∂t = -Δu + f(u) and u(0, ·) = θ{symbol}(·), where u is defined on [0,T)×RN and f is a nonlinear complex-valued function. We consider the case of the critical power. That is, f(z) ≅ - |z| 4 N z as z → + ∞ in a sense which will be made precise. In addition, we suppose that u(t) blows up in the norm ∥·∥H at time T. We prove that u(t) has no limit in L2 as t → T. In the particular case where θ{symbol} has a spherical symmetry, we further show a phenomenon of L2 concentration at the origin. © 1990.