A Sharp Threshold Of Blow-up For An Inhomogeneous Nonlinear Schr(?)dinger Equation

The aim of this paper is to study the Cauchy problem of the following inhomogeneous nonlinear Schr(?)dinger equation(?)where φ=φ(t,x): R×Rn→C, b∈(0, min{2, n}),1+4-2b/n<p < 1+4-2b/n-2.Firstly, we obtain the existence of blow-up solutions with positive energy. Sec-ondly, we establish a sharp threshold of global existence and blow-up for the case(?) (α is a given constant, sc =n/2-2-b/p-1), where M[φ] and E[φ] denote the mass and energy of φ, respectively, and Q is the ground state solution to -△Q+Q-|x|-b|Q|p-1Q=0. This result extends the conclusion of Farah [22] (J. Evol. Eq, 2016) where a sharp threshold of blow-up is obtained for the complementary case M[φ]1-sc/sc E[φ] < M[Q]1-sc/sc E[Q].