Existence of Positive Solutions for a Critical Nonlocal Elliptic System

We establish the existence of positive solution to the critical nonlocal elliptic system $$(S)\hskip10mm \left\{ \begin{aligned} & (-\Delta)^{s}_p u+a(x)|u|^{p-2} u+ c(x) |v|^{p-2} v = \tfrac{1}{p^{*}_s}K_u(u,v) \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ & (-\Delta)^{s}_p v+c(x)| u|^{p-2} u+ b(x)|v|^{p-2} v = \tfrac{1}{p^{*}_s}K_v(u,v) \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ &\ u, v>0 \ \mbox{in} \ \mathbb{R}^{N},\ u, v \in D^{s, p}(\mathbb{R}^{N}),\ N> ps,\ s\in (0,1). \end{aligned} \right.$$ Here $(-\Delta)^{s}_p$ denotes the fractional $p$\,-Laplacian, $a,b$ and $c$ are suitable functions and $K$ is a $p^{*}_s$-homogeneous function, $p^{*}_s= (pN)/(N-ps)$, $N > ps$. One of the main tools is to apply the global compactness result for the associated energy functional similar to that due to M.\,Struwe [{\it A global compactness result for elliptic boundary value problems involving limiting nonliarities}, Math. Zeitschrift 187/4 (1984) 511--517] combined with some information on a limit system of $(S)$ with $a=b=c=0$, the concentration compactness due to P.\,L.\,Lions [{\it The concentration-compactness principle in the calculus of variations. I: The limit case}, Rev. Mat. Iberoamericana 1/1 (1985) 145--201] and the Brouwer degree theory.