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

For a competition-diffusion system involving the fractional Laplacian of the form

$?{(?\mathrm{\Delta })}^{s}u=u{v}^2,?{(?\mathrm{\Delta })}^{s}v=v{u}^2,u,v>0\text{in}{\mathbb{R}}^N,$

with $s\in (0,1)$, 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 $N\ge 3$. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.