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

For a class of competition-diffusion nonlinear systems involving the square root of the laplacian, including the fractional Gross-Pitaevskii system. (-Delta)(1/2)ui = omega(!)u(i)(3) + lambda(i)u(i) - beta u(i) [GRAPHICS] a(ij)u(j)(2), i = 1, ... , k, we prove that L-infinity boundedness implies C-0,C-alpha boundedness for every alpha epsilon [0, 1/ 2), uniformly as beta -> infinity. Moreover we prove that the limiting profile is C-0,C-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.