Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

Abstract

For the positive solutions of the Gross-Pitaevskii systemwe prove that L ∞ -boundedness implies C 0,α -boundedness, uniformly as β → +∞, for every α ∈ (0, 1). Moreover we prove that the limiting profile, as β → +∞, is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given.MSC : 35B40, 35B45, 35J55.

Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent

“…It is expected that components of the limiting profile tend to separate in different regions of the underlying domain, . This phenomenon, called phase separation, has been well studied for L ∞ -bounded positive solutions of system (1.2) in the case N = 2, 3 by Noris et al [27] and Wei and Weth [33,34]. For other kinds of elliptic systems with strong competition, phase separation has also been well studied; we refer to [13][14][15] and references therein.…”

Nodal set of strongly competition systems with fractional Laplacian

“…Regarding the standard Laplace diffusion case, abroad literature is present, starting from [5,[9][10][11][12]16], in a series of recent papers [2,4,[21][22][23]28,30,[32][33][34], also in the parabolic case [13][14][15]29]. Among the others, the following results are known: the uniform in k bounds for solutions of corresponding systems in Hölder spaces, the regularity of the limiting profiles and the regularity of the free boundaries in the singular limit.…”

Rotating multicomponent Bose–Einstein condensates

“…At temperatures T much smaller than the critical temperature T c [34], for a k-component BEC, its wave function can be well described by the self-consistent nonlinear Schröding-er equations [10,[24][25][26][30][31][32][33][34][35][36][37][38], know as coupled Gross-Pitaevskii equations [6,38,39]:…”

“…For solitary wave solutions of the form u j (x, t) = e iλj t ψ j (x)(j = 1, 2) and Ω = 0 of system (1.1), one investigates the phase separation phenomena [10,11,33,40,37]. Where ψ 1 (x) and ψ 2 (x) are real functions, which satisfy the elliptic system.…”

“…From a rigorous mathematical point of view, the phase separation is not well understood so far for the rotating k-mixture BEC (1.4). The method in [10,11,33,37,40] can not be applied to the this case. The main purpose of this paper is to study the spatial behavior of the minimizer of the functional (1.5).…”