This article concerns the diffusion system
\displaylines{
\partial_tu-\Delta_{x}u+V(x)u=g(t,x,v),\cr
-\partial_tv-\Delta_{x}v+V(x)v=f(t,x,u),
}
where z=(u,v):RΓRNβR2,
V(x)βC(RN,R) is a general periodic function, g, f
are periodic in t, x and asymptotically linear in u, v at infinity.
We find a minimizing Cerami sequence of the energy functional outside
the Nehari-Pankov manifold N and therefore obtain ground
state solutions