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

Abstract

ABSTRACTWe consider in the whole plane the following Hamiltonian coupling of Schrödinger equationswhere 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 up to translations. Moreover, for each (u,v)∈S, one has that u,v are uniformly bounded in 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 systemwhere V∈𝒞(ℝ2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant 𝜀→0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.