Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian

Source abstract

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross–Pitaevskii system (-\Delta)^{1/2} u_i=\omega_i u_i^3 + \lambda_i u_i - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, we prove that L^\infty boundedness implies \mathcal C^{0,\alpha} boundedness for every \alpha\in[0,1/2) , uniformly as \beta\to +\infty . Moreover we prove that the limiting profile is \mathcal C^{0,1/2} .This system arises, for instance, in the relativistic Hartree—Fock approximation theory for k -mixtures of Bose–Einstein condensates in different hyperfine states.