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

TL;DR

This paper studies ground state solutions of coupled critical Schrödinger equations in two dimensions, establishing their compactness, positivity, symmetry, and existence of semiclassical states concentrating near potential minima.

Contribution

It proves the compactness, positivity, and radial symmetry of ground states, and demonstrates the existence of semiclassical solutions concentrating at potential minima as the Planck constant approaches zero.

Findings

Ground states are compact up to translations.

Ground states are positive and radially symmetric.

Existence of semiclassical solutions concentrating at potential minima.

Abstract

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.