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Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. (English) Zbl 1189.35314

Summary: For the positive solutions of the Gross-Pitaevskii system
\[ \begin{aligned} -\Delta u_\beta+\lambda_\beta u_\beta&= \omega_1u_\beta^3-\beta u_\beta v_\beta^2,\\ -\Delta v_\beta+\mu_\beta v_\beta&= \omega_2v_\beta^3-\beta u_\beta^2 v_\beta, \end{aligned} \]
we prove that \(L^\infty\)-boundedness implies \(C^{0,\alpha}\)-boundedness for every \(\alpha\in(0,1)\), uniformly as \(\beta\to+\infty\). Moreover, we prove that the limiting profile as \(\beta\to+\infty\) 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

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
81V80 Quantum optics
35B45 A priori estimates in context of PDEs
35B09 Positive solutions to PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
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References:

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