On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition

Abstract

For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-Δ)^su=uv^2,\quad-(-Δ)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \ \mathbb{R}^N, \end{equation*} whith $s\in (0,1)$, 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 $N\ge 3$.

31 pages

References in corpus (4)

• Uniform bounds for strongly competing systems: the optimal Lipschitz case
• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation
• On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects
• Multidimensional entire solutions for an elliptic system modelling phase separation

Cited by in corpus (3)

• On s-harmonic functions on cones
• Regularity of shape optimizers for some spectral fractional problems
• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory