Ground states for a linearly coupled system of Schr\"odinger equations on ${\mathbb{R}}^N$

We study the following class of linearly coupled Schr"{o}dinger elliptic systems $$\{\begin{array}{lr}-\mathrm{\Delta }u+V_1(x)u=\mu \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\lambda (x)v,& x\in {\mathbb{R}}^N,\\ -\mathrm{\Delta }v+V_2(x)v=\mathrm{\mid }v{\mathrm{\mid }}^{q-2}v+\lambda (x)u,& x\in {\mathbb{R}}^N,\end{array}$$ where $N\ge 3$, $2<p\le q\le 2^{\ast }=2N\mathrm{/}(N-2)$ and $\mu \ge 0$. We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term $\lambda (x)$ by the assumption $\mathrm{\mid }\lambda (x)\mathrm{\mid }\le \delta \sqrt{V_1(x)V_2(x)}$, for some $0<\delta <1$. We deal with three cases: Firstly, we study the subcritical case, $2<p\le q<2^{\ast }$, and we prove the existence of positive ground state for all parameter $\mu \ge 0$. Secondly, we consider the critical case, $2<p<q=2^{\ast }$, and we prove that there exists ${\mu }_0>0$ such that the coupled system possesses positive ground state solution for all $\mu \ge {\mu }_0$. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case $p=q=2^{\ast }$, and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.