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−(−Δ)su=uv2,−(−Δ)sv=vu2,u,v>0inRN, with 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.