In this paper, we consider a class of focusing nonlinear Schrödinger equations involving power-type nonlinearity with critical Sobolev exponent{(i∂∂t+Δ)u+|u|4N−2su=0,inR+×RN,u=u0(x)∈H1(RN)∩L2(RN,|x|2dx),fort=0,x∈RN, where 2N−2s=:p⋆ be the critical Sobolev exponent with s<N2. For dimension N≥1, the initial data u0 belongs to the energy space and |⋅|u0∈L2(RN) and the power index s satisfies 0,1∋s≡sc=N2−1p⋆, we prove that the problem is non-global existence in H1(RN) (here, finite-time blow-up occurs) with the energy of initial data Eu0 is negative. Moreover, we establish the stability for the solutions with the lower bound and the global a priori upper bound in dimension N≥2 related conservation laws. The motivation for this paper is inspired by the mass critical case with s=0 of the celebrated result of B. Dodson 9 and the work of R. Killip and M. Visan 18 represented with energy critical case for s=1. Our new results mention to nonlinear Schrödinger equation for interpolating between mass critical or mass super-critical (s≥0) and energy sub-critical or energy critical (s≤1).