  We consider in the whole plane the Hamiltonian coupling of semilinear
Schroedinger equations which have critical growth in the sense of Moser. We
prove that the (nonempty) set S of ground state solutions is compact up to
translations. Moreover, ground states are uniformly bounded and uniformly
decaying at infinity. Then we prove that actually the ground state is positive
and radially symmetric. We apply those results to prove the existence of
semiclassical ground states solutions to the singularly perturbed system when
the external potential stays bounded away from zero. Namely, as the
adimensionalized Planck constant vanishes, we prove the existence of minimal
energy solutions which concentrate around the closest local minima of the
potential with some precise asymptotic rate.

Analysis of PDEs