Ground states for a linearly coupled system of Schrodinger equations on RN

We study the following class of linearly coupled Schrodinger elliptic systems [GRAPHICS] where N >= 3, 2 < p <= q <= 2* = 2N/(N - 2) and mu >= 0. We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term lambda(x) by the assumption |lambda(x)| <= delta root V1(x)V-2(x), for some 0 < delta < 1. We deal with three cases: Firstly, we study the subcritical case, 2 < p <= q < 2*, and we prove the existence of positive ground state for all parameter mu >= 0. 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 >= 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.