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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 | 4 N − 2 s u = 0 , in R + × R N , u = u 0 (x) ∈ H 1 (R N) ∩ L 2 (R N , | x | 2 d x) , for t = 0 , x ∈ R N , where 2 N − 2 s = : p ⋆ be the critical Sobolev exponent with s < N 2. For dimension N ≥ 1 , the initial data u 0 belongs to the energy space and | ⋅ | u 0 ∈ L 2 (R N) and the power index s satisfies [ 0 , 1 ] ∋ s ≡ s c = N 2 − 1 p ⋆ , we prove that the problem is non-global existence in H 1 (R N) (here, finite-time blow-up occurs) with the energy of initial data E [ u 0 ] 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). [ABSTRACT FROM AUTHOR]

In this paper, we simplify the proof of M. Hamano in [13] , scattering theory of the solution to (NLS system) , by using the method from B. Dodson and J. Murphy in [10]. Firstly, we establish a criterion to ensure the solution scatters in H 1 (R 5) × H 1 (R 5). In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concentrated. [ABSTRACT FROM AUTHOR]