  We study the existence of positive solutions for the following class of
$(p,q)$-Laplacian coupled systems
  \[
  \left\{
  \begin{array}{lr}
  -\Delta_{p} u+a(x)|u|^{p-2}u=f(u)+
\alpha\lambda(x)|u|^{\alpha-2}u|v|^{\beta}, & x\in\mathbb{R}^{N},
  -\Delta_{q} v+b(x)|v|^{q-2}v=g(v)+ \beta\lambda(x)|v|^{\beta-2}v|u|^{\alpha},
& x\in\mathbb{R}^{N},
  \end{array}
  \right.
  \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $\lambda(x)$ of
the coupling term is related with the potentials by the condition
$|\lambda(x)|\leq\delta a(x)^{\alpha/p}b(x)^{\beta/q}$ where $\delta\in(0,1)$
and $\alpha/p+\beta/q=1$. We deal with periodic and asymptotically periodic
potentials. The nonlinear terms $f(s), \; g(s)$ are "superlinear" at $0$ and at
$\infty$ and are assumed without the well known Ambrosetti-Rabinowitz condition
at infinity. Thus, we have established the existence of positive ground states
solutions for a large class of nonlinear terms and potentials. Our approach is
variational and based on minimization technique over the Nehari manifold.
