Dynamic of threshold solutions for energy-critical NLS

We consider the radial energy-critical non-linear focusing Schr"odinger equation in dimension N=3,4,5. An explicit stationnary solution, W, of this equation is known. In a previous work by C. Carlos and F. Merle, the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u)=E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.