  We study the following class of linearly coupled Schr\"{o}dinger elliptic
systems $$\left\{ \begin{array}{lr} -\Delta
u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta
v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right.
$$ where $N\geq3$, $2<p\leq q\leq 2^{*}=2N/(N-2)$ and $\mu\geq0$. We consider
nonnegative potentials periodic or asymptotically periodic which are related
with the coupling term $\lambda(x)$ by the assumption
$|\lambda(x)|\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$, for some $0<\delta<1$. We deal
with three cases: Firstly, we study the subcritical case, $2<p\leq q<2^{*}$,
and we prove the existence of positive ground state for all parameter
$\mu\geq0$. Secondly, we consider the critical case, $2<p<q=2^{*}$, and we
prove that there exists $\mu_{0}>0$ such that the coupled system possesses
positive ground state solution for all $\mu\geq\mu_{0}$. In these cases, we use
a minimization method based on Nehari manifold. Finally, we consider the case
$p=q=2^{*}$, and we prove that the coupled system has no positive solutions.
For that matter, we use a Pohozaev identity type.
