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

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