Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

Abstract

In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.

Keywords

• inhomogeneous nonlinear schrödinger equation
• inverse-square potential
• blow-up
• global existence
• mass concentration