Uniform H\"older bounds for nonlinear Schr\"odinger systems with strong competition

TL;DR

This paper proves that positive solutions to a competitive nonlinear Schr"odinger system become uniformly H"older continuous as the competition parameter increases, with the limiting profile being Lipschitz continuous, using blow-up techniques and monotonicity formulas.

Contribution

It establishes uniform H"older bounds for solutions of the Gross-Pitaevskii system with strong competition, extending results to systems with multiple densities.

Findings

L^0 boundedness implies uniform Hf6lder boundedness as competition increases

Limiting profiles are Lipschitz continuous

Extensions to systems with more than two densities

Abstract

For the positive solutions of the competitive Gross-Pitaevskii system of two equations, we prove that L^\infty boundedness implies uniform H\"older boundedness as the competition parameter goes to infinity. Moreover we prove that the limiting profile is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with more than two densities are given.