Symmetry and monotonicity of positive solutions for Choquard equations involving a generalized tempered fractional p-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 a generalized tempered fractional p-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-λf)psu(x)+ωu(x)=Cn,t(|x|2t-n∗uq)uq-1,x∈Rn,u(x)>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}$$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta -\lambda _{f})_{p}^{s}u(x)+\omega u(x)=C_{n,t}(|x|^{2t-n}*u^{q})u^{q-1}, &{}x\in {\mathbb {R}}^{n},\\ u(x)>0,&{}x\in {\mathbb {R}}^{n}, \end{array} \right. \end{aligned}$$\end{document}where 0<s,t<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s,\ t<1$$\end{document}, p>2,p-1<q<∞,n≥2,ω>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2,\ p-1<q<\infty ,\ n\ge 2,\ \omega >0$$\end{document}. By using the direct method of moving planes, we prove that the positive solutions of system above must be radially symmetric and monotone decreasing about some point in the whole space. In particular, decay at infinity and narrow region principle play an important role in getting the main results.