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

Abstract

Abstract. For a competition-diffusion system involving the fractional Laplacian of the formwhith s ∈ (0, 1), we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when N = 2 with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when N ≥ 3. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.

On s-harmonic functions on cones

“…For the case of standard diffusion, both the asymptotic analysis and the properties of the segregated limiting profiles are fairly well understood, we refer to [13,15,16,24,28] and references therein. Instead, when the diffusion is nonlocal and modeled by the fractional Laplacian, the only known results are contained in [29,30,31,32]. As shown in [29,30], estimates in Hölder spaces can be obtained by the use of fractional versions of the Alt-Caffarelli-Friedman (ACF) and Almgren monotonicity formulae.…”

“…It is easy to see, by a Schwarz symmetrization argument, that ν ACF s is achieved by a pair of complementary spherical caps (ω θ , ω π−θ ) ∈ P 2 with aperture 2θ and θ ∈ (0, π) (for a detailed proof of this kind of symmetrization we refer to [31]), that is:…”