Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

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Abstract
For the positive solutions of the Gross–Pitaevskii system 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.
Funding Information
  • MIUR, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
  • Fundação para a Ciência e a Tecnologia (SFRH/ BD/28964/2006, Financiamento Base 2008 - ISFL/1/209)
  • Fundação Calouste Gulbenkian (“Estímulo à Investigação 2007”)

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