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

Abstract

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross–Pitaevskii system

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.