Abstract

We establish the existence of positive solution to the critical nonlocal elliptic system (S){(−Δ)psu+a(x)∣u∣p−2u+c(x)∣v∣p−2v=1ps∗Ku(u,v)  in  RN,(−Δ)psv+c(x)∣u∣p−2u+b(x)∣v∣p−2v=1ps∗Kv(u,v)  in  RN, u,v>0 in RN, u,v∈Ds,p(RN), N>ps, s∈(0,1).(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) \ \ \text{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) \ \ \text{in} \ \ \mathbb{R}^{N},\\ &\ u, v>0 \ \text{in} \ \mathbb{R}^{N},\ u, v \in D^{s, p}(\mathbb{R}^{N}),\ N> ps,\ s\in (0,1). \end{aligned} \right. Here (−Δ)ps(-\Delta)^{s}_p denotes the fractional pp -Laplacian, a,ba,b and cc are suitable functions and KK is a ps∗p^{*}_s-homogeneous function, ps∗=(pN)/(N−ps)p^{*}_s= (pN)/(N-ps), N>psN > 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 [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)(S) with a=b=c=0a=b=c=0, the concentration compactness due to P. L. Lions [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.

Author information

Contact details are reproduced from the original publication and may be historical.

Augusto C. R. Costa

Inst. de Ciências Exatas e Naturais, Faculdade de Matemática, Universidade Federal do Pará, Belém, Brazil

aug@ufpa.br

Giovany M. Figueiredo

Dep. de Matemática, Universidade de Brasilia, Brazil

giovany@unb.br

Olimpio H. Miyagaki

Dep. de Matemática, Universidade Federal de São Carlos, Brazil

olimpio@ufscar.br

Keywords

  • Variational critical system
  • fractional equations
  • global compactness result
  • Brouwer degree theory

Mathematics Subject Classification

35J50, 35R11; 58E05, 47H11

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Suggested citation

A. C. R. Costa, G. M. Figueiredo, O. H. Miyagaki. “Existence of Positive Solutions for a Critical Nonlocal Elliptic System.” Journal of Convex Analysis 29 (2022), No. 4, 1083–1117. https://doi.org/10.68381/jca29061

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