Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schrödinger equation with inverse-square potential

Abstract

We consider the $L^2$-critical NLS with inverse-square potential $$i{\partial }_{t}u+\Delta u+c|x{|}^{-2}u=-|u{|}^{\frac{4}{d}}u,u(0)=u_0,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^d,$$ where $d\ge 3$ and $c\ne 0$ satisfies $c<\lambda (d):={\left(\frac{d-2}{2}\right)}^2$. Using a refined compactness lemma, we extend the mass concentration of finite time blow-up solutions established in the attractive case by the first author in [Bensouilah] to $c<\lambda (d)$. By means of a simple and short limiting profile theorem, we get the same classification result obtained by Csobo and Genoud in [CsoboGenoud] for $0<c<\lambda (d)$. It also enables us to extend the classification to $c<\lambda (d)$.

16 pages, extended version with more details