A priori estimates and positivity for semiclassical ground states for   systems of critical Schroedinger equations in dimension two
  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. 
Daniele Cassani and Jianjun Zhang
math.AP