Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition



For the positive solutions of the Gross–Pitaevskii system

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we prove that L-boundedness implies C0,α-boundedness for every α ϵ (0,1), uniformly as β → +∞. Moreover, we prove that the limiting profile as β → +∞ is Lipschitz-continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and 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 k > 2 densities are given. © 2009 Wiley Periodicals, Inc.