A priori estimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations (Formula presented.) where V0>0, f,g have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground-state solutions is compact in H1(ℝ2) × H1(ℝ2) up to translations. Moreover, for each (u,v)∈S, one has that u,v are uniformly bounded in L∞(ℝ2) 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-state solutions to the singularly perturbed system (Formula presented.) where V∈(ℝ2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant →0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.