Global existence and blow-up for the variable coefficient Schrödinger equations with a linear potential

Abstract

In this paper, we study a class of variable coefficient Schrödinger equations with a linear potential \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,\] where $2-n<b\le 0,c\ge b-2$ and $0<{\text{p}}_c\le (2-b)(p+2)$, where ${\text{p}}_c:=np-2c$. In the radial or finite variance case, we firstly prove the global existence and blow-up below the ground state threshold for the mass-critical and inter-critical nonlinearities. Next, adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a sufficient condition on the nonradial initial data, under which the global behavior of the general solution is established.

39 pages