  For a class of competition-diffusion nonlinear systems involving the square
root of the Laplacian, including the fractional Gross-Pitaevskii system, we
prove that uniform boundedness implies Holder boundedness for every exponent
less than 1/2, uniformly as the interspecific competition parameter diverges.
Moreover we prove that the limiting profile is Holder continuous of exponent
1/2. This system arises, for instance, in the relativistic Hartree-Fock
approximation theory for mixtures of Bose-Einstein condensates in different
hyperfine states.

Analysis of PDEs