  In this paper, we consider the strongly coupled nonlinear Kirchhoff-type
system with vanshing potentials: \begin{equation*}\begin{cases}
-\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\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|^2\dx\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}(\R^3), \end{cases}\end{equation*} where $a_i>0$ are
constants, $\lambda,b_i>0$ are parameters for $i=1,2$, $\alpha,\beta>1$ and
$\alpha+\beta\leqslant 4$, $V(x)$, $W(x)$ are nonnegative continuous
potentials, the nonlinear term $F(x,u,v)=|u|^\alpha|v|^\beta$ is not
4-superlinear at infinity. Such problem cannot be studied directly by standard
variational methods, even by restricting the associated energy functional on
the Nehari manifold, because Palais-Smale sequences may not be bounded.
Combining some new detailed estimates with 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 $4<\alpha+\beta<6$.
