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 systemwe prove that L ∞ boundedness implies C 0,α boundedness for every α ∈ [0, 1/2), uniformly as β → +∞. Moreover we prove that the limiting profile is 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. thus focusing on the singular limit problem obtained when the (positive) parameter β, accounting for the competitive interactions, diverges to ∞. Among the others, the cases f i (s) = r i s(1−s/K i ), g ij (s) = a ij s (logistic internal dynamics with Lotka-Volterra competition) and f i (s) = ω i s 3 + λ i s, g ij (s) = a ij s 2 (focusing-defocusing 2010 Mathematics Subject Classification. Primary: 35J65; secondary: 35B40 35B44 35R11 81Q05 82B10.

Variational Problems with Long-Range Interaction

“…belong to the same functional class [13,25], and hence in particular satisfy the same free-boundary condition, that is |∂ ν u i (x 0 )| = |∂ ν u j (x 0 )| on the regular part of the free boundary. A similar difference has been observed in [27,28,29] in the case of fractional operators, that is when the non-locality is in the differential operator.…”

Nodal set of strongly competition systems with fractional Laplacian

“…Verzini and Zilio [27] proved the local uniform C α (0 < α < 1) bounds for solutions to system (1.1) (similar results concerning the systems of the relativistic version of Bose-Einstein condensation were given in [25,26], to which we refer for further details). In our recent paper [31], we gave a local uniform Lipschitz bound, using Kato inequality.…”

On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition

“…Competition-diffusion nonlinear systems with k-components involving the fractional Laplacian have been the object of a recent literature, starting with [19,20], where the authors provided asymptotic estimates for solutions to systems of the form…”