  We investigate the existence and multiplicity of weak solutions for a
nonlinear Kirchhoff type quasilinear elliptic system on the whole space
$\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally
super-$(m_1,m_2)$ condition, that is,
$\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty
\mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is
weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally
global restriction,
$\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty
\mbox{ for a.e. } x \in \mathbb{R}^N$. We obtain that system has at least one
weak solution by using the classical Mountain Pass Theorem. To a certain
extent, our theorems extend the results of Tang-Lin-Yu [Journal of Dynamics and
Differential Equations, 2019, 31(1): 369-383]. Moreover, under the above
naturally global restriction, we obtain that system has infinitely many weak
solutions of high energy by using the Symmetric Mountain Pass Theorem, which is
different from those results of Wang-Zhang-Fang [Journal of Nonlinear Sciences
and Applications, 2017, 10(7): 3792-3814] even if we consider the system on the
bounded domain with Dirichlet boundary condition.
