Global existence and blow-up for the variable coefficient Schr\"{o}dinger equations with a linear potential

In this paper, we study a class of variable coefficient Schr\"{o}dinger equations with a linear potential $$i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,$$ where $2-n<b\leq0,\ c\geq b-2$ and $0<\textbf{p}_c\leq(2-b)(p+2)$, where $\textbf{p}_c:=np-2c$. 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.