  In this paper, we investigate the existence of solutions for a class of
quasilinear elliptic system
  \begin{eqnarray*}
  \begin{cases}{ccc}
  -\mbox{div}(\phi_1(|\nabla u|)\nabla u)+V_1(x)\phi_1(|u|)u=\lambda F_u(x,
u,v), \ \ x\in \mathbb R^N,
  -\mbox{div}(\phi_2(|\nabla v|)\nabla v)+V_2(x)\phi_2(|v|)v=\lambda F_v(x,
u,v), \ \ x\in \mathbb R^N,
  u\in W^{1,\Phi_1}(\mathbb R^N), v\in W^{1,\Phi_2}(\mathbb R^N),
  \end{cases}
  \end{eqnarray*} where $N\ge 2$, $\inf_{\mathbb R^N}V_i(x)>0,i=1,2$, and
$\lambda>0$. We obtain that when the nonlinear term $F$ satisfies some growth
conditions only in a circle with center $0$ and radius $4$, system has a
nontrivial solution $(u_\lambda,v_\lambda)$ with
$\|(u_{\lambda},v_{\lambda})\|_{\infty}\le 2$ for every $\lambda$ large enough,
and the families of solutions $\{(u_\lambda,v_\lambda)\}$ satisfy that
$\|(u_\lambda,v_\lambda)\|\to 0$ as $\lambda\to \infty$. Moreover, a
corresponding result for a quasilinear elliptic equation is also obtained,
which is better than the result for the elliptic system.
