An asymptotic analysis and stability for a class of focusing Sobolev critical nonlinear Schr?dinger equations

In this paper, we consider a class of focusing nonlinear Schrodinger equations involving power-type nonlinearity with critical Sobolev exponent ⠂(i partial differential partial differential t+⠃)u+ |u|4 N-2s u = 0, in R+ x RN, u = u0(x) e H1(RN) n L2(RN, |x|2dx), for t = 0, x e RN, where 2 N-2s =: p ⠄ be the critical Sobolev exponent with s < N2 . For dimension N > 1, the initial data u0 belongs to the energy space and | center dot |u0 e L2(RN) and the power index ssatisfies [0, 1] ⠆s= sc= 2N - p ⠄1, we prove that the problem is non-global existence in H1(RN) (here, finite-time blow-up occurs) with the energy of initial data E[u0] 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 Schrodinger equation for interpolating between mass critical or mass super-critical (s > 0) and energy sub-critical or energy critical (s < 1).(c) 2023 Elsevier Inc. All rights reserved.