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Journal of Convex Analysis 29 (2022), No. 4, 1083--1117
Copyright Heldermann Verlag 2022

Existence of Positive Solutions for a Critical Nonlocal Elliptic System

Augusto C. R. Costa
Inst. de Ciencias Exatas e Naturais, Faculdade de Matemática, Universidade Federal do Pará, Belém, Brazil

Giovany M. Figueiredo
Dep. de Matemática, Universidade de Brasilia, Brazil

Olimpio H. Miyagaki
Dep. de Matemática, Universidade Federal de Sao Carlos, Brazil


We establish the existence of positive solution to the critical nonlocal elliptic system \begin{equation*} (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. \end{equation*} 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.

Keywords: Variational critical system, fractional equations, global compactness result, Brouwer degree theory.

MSC: 35J50, 35R11; 58E05, 47H11.

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