Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition
Benedetta Noris, Susanna Terracini, Hugo Tavares, Gianmaria Verzini
Source abstract
Abstract For the positive solutions of the Gross–Pitaevskii system we prove that L ∞ ‐boundedness implies C 0,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.
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