Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation

International audienceIn this paper, we consider the nonlinear Schrödinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein \cite{MR84d:35140,MR87i:35026}. Furthermore, we improve a blow-up condition