  In this paper, we consider the strongly coupled nonlinear Kirchhoff-type
system with vanshing potentials: \[\begin{cases}
-\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+\lambda
V(x)u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\mathbb{R}^3,\\
-\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2dx\right)\Delta v+\lambda
W(x)v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\mathbb{R}^3,\\
u,v\in \mathcal{D}^{1,2}(\mathbb{R}^3), \end{cases}\] where $a_i>0$ are
constants, $\lambda,b_i>0$ are parameters for $i=1,2$, $\alpha,\beta>1$ satisfy
$\alpha+\beta\le4$, the nonlinear term $F(x,u,v)=|u|^\alpha|v|^\beta$ is not
4-superlinear at infinity, $V(x)$, $W(x)$ are nonnegative continuous
potentials. By establishing some new estimates and truncation technique, we
obtain the existence of positive vector solutions for the above system when
$b_1+b_2$ small and $\lambda$ large. Moreover, the asymptotic behavior of these
vector solutions is also explored as $\textbf{b}=(b_1,b_2)\to \bf{0}$ and
$\lambda\to\infty$. In particular, our results extend some known ones in
previous papers that only deals with the case where $\alpha,\beta>2$ with
$\alpha+\beta<6$.
