Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition

Abstract

For the positive solutions of the competitive Gross-Pitaevskii system of two equations, we prove that L^\infty boundedness implies uniform Hölder 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.

27 pages

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