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

TL;DR

This paper establishes uniform Holder bounds for strongly competing systems involving the square root of the Laplacian, including fractional Gross-Pitaevskii systems, showing boundedness implies Holder continuity with exponent less than 1/2.

Contribution

It proves uniform Holder bounds and describes the limiting profile's regularity for a class of nonlinear systems involving fractional Laplacians, extending previous regularity results.

Findings

Uniform boundedness implies Holder boundedness for exponents less than 1/2.

The limiting profile is Holder continuous with exponent 1/2.

Results apply to systems arising in Bose-Einstein condensate models.

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 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.