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

In this paper, we study a class of variable coefficient Schrödinger equations with a linear potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\nabla }\cdot (\mathrm{\mid }x{\mathrm{\mid }}^b\mathrm{\nabla }u)-V(x)u=-\mathrm{\mid }x{\mathrm{\mid }}^c\mathrm{\mid }u{\mathrm{\mid }}^{p}u,$$ where $2-n<b\le 0,c\ge b-2$ and $0<{\mathbf{\text{p}}}_c\le (2-b)(p+2)$, where ${\mathbf{\text{p}}}_c:=np-2c$. 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.