  In this paper, we deal with the following Cauchy problem \begin{equation*}
\left\{ \begin{array}{lll} iu_t = \Delta u + 2uh'(|u|^2)\Delta h(|u|^2) +
V(x)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}^N.
\end{array}\right. \end{equation*} Here $h(s)$ and $V(x)$ are some real
functions. We take the potential $V(x)\in
L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ as criterion of the blowup and
global existence of the solution to (1.1). In some cases, we can classify it in
the following sense: If $V(x)\in S(I)$, then the solution of (1.1) is always
global existence for any $u_0$ satisfying $0<E(u_0)<+\infty$; If $V(x)\in
S(II)$, then the solution of (1.1) may blow up for some initial data $u_0$.
Here $$ S(I)=\cup_{q>q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)],\quad
S(II)=\left\{\cup_{q<q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)]\right\}\setminus
S(I).$$ Under certain assumptions, we also establish Morawetz estimates and
spacetime bounds for the global solution, for example, \begin{align*}
&\int_0^{+\infty} \int_{\mathbb{R}^N }\frac{[|\nabla h(|u|^2)|^2 +
|V(x)||u|^2]}{(|x|+t)^{\lambda}}dxdt\leq C,\\ & \|u\|_{L^{\bar{q}}_t
(\mathbb{R}) L^{\bar{r}}_x(\mathbb{R}^N)} = \left(\int_0^{+\infty}
\left(\int_{\mathbb{R}^N}|u|^{\bar{r}} dx\right)^{\frac{\bar{q}}{\bar{r}}}
dt\right)^{\frac{1}{\bar{q}}} \leq C. \end{align*}
