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

TL;DR

This paper investigates the growth behavior of positive solutions to a fractional Laplacian competition system, establishing the maximal growth rate as 2s, constructing solutions near this bound, and demonstrating higher-dimensional solutions.

Contribution

It proves the asymptotic growth rate for solutions is 2s, constructs near-critical solutions in two dimensions, and shows existence of higher-dimensional solutions.

Findings

Maximal growth rate for solutions is 2s.

Constructed symmetric solutions with growth close to the maximum.

Established existence of higher-dimensional solutions for N ≥ 3.

Abstract

For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-\Delta)^su=uv^2,\quad-(-\Delta)^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 $N\geq 3$.