Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations

In this paper, we consider the following pseudo-relativistic Choquard equations: (-Δ+m2)su+wu=RN,t1|x-y|N-2t∗upuq,inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta +m^{2})^{s} u+wu=R_{N,t}\left( \frac{1}{|x-y|^{N-2t}}*u^{p}\right) u^{q}, \quad \mathrm{in} \;\;\mathbb {R}^{N}, \end{aligned}$$\end{document}where s,t∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s,t\in (0,1)$$\end{document}, mass m>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>0$$\end{document}, w>-m2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w>-m^{2s}$$\end{document}, 2<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<\infty $$\end{document}, and 0<q≤p-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q\le p-1$$\end{document}. We first establish a narrow region principle for pseudo-relativistic Choquard equations and estimate the decay of the solutions at infinity. Using the generalized direct method of moving planes, we obtain the radial symmetry and monotonicity of nonnegative solutions for the above equations.