Existence of solutions for a quasilinear elliptic system with local nonlinearity on $\mathbb R^N$

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.