Symmetry and monotonicity of positive solutions for a system involving fractional p&q-Laplacian in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document}

In this paper, we study a nonlinear system involving the fractional p&q-Laplacian in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document}(-Δ)ps1u(x)+(-Δ)qs2u(x)=f(u(x),v(x)),x∈Rn,(-Δ)ps1v(x)+(-Δ)qs2v(x)=g(u(x),v(x)),x∈Rn,u,v>0,x∈Rn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )_{p}^{s_{1}}u(x)+(-\Delta )_{q}^{s_{2}}u(x)=f(u(x),v(x)), &{}x\in {\mathbb {R}}^{n},\\ (-\Delta )_{p}^{s_{1}}v(x)+(-\Delta )_{q}^{s_{2}}v(x)=g(u(x),v(x)), &{}x\in {\mathbb {R}}^{n},\\ u,v>0,&{}x\in {\mathbb {R}}^{n}. \end{array} \right. \end{aligned}$$\end{document}where 0<s1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s_{1}$$\end{document}, s2<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{2}<1$$\end{document}, p,q>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q>2$$\end{document}. By using the direct method of moving planes, we prove that the positive solution (u, v) of system above must be radially symmetric and monotone decreasing in the whole space.