Necessary conditions and sufficient conditions for global existence in the nonlinear Schr\"odinger equation

In this paper, we consider the nonlinear Schr\"odinger 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. Furthermore, we improve a blow-up condition.