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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]

We study the focusing nonlinear Schrödinger equation with inverse-square potential i ∂ t u + Δ u − a | x | 2 u + | u | p − 1 u = 0 with a > − 1 4 and 3 4 < p − 1 < 4 in three dimensions. In this paper we obtain blow-up and global existence criteria for the solution to ( N L S a ) by the method from Duyckaerts-Roudenko [1], which extend a recent work of Killip-Murphy-Visan-Zheng [2] to above the threshold. [ABSTRACT FROM AUTHOR]

We study the focusing inhomogeneous nonlinear Schrödinger equation with inverse‐square potential i∂tu+Δu−a|x|2u+|x|−b|u|2u=0,$$ i{\partial}_tu&amp;#x0002B;\Delta u-\frac{a}{{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;2}u&amp;#x0002B;{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;{-b}{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2u&amp;#x0003D;0, $$where a>−14$$ a&gt;-\frac{1}{4} $$ and 0