Bounded Engel elements in residually finite groups

Let q be a prime. Let G be a residually finite group satisfying an identity. Suppose that for every x∈G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in G$$\end{document} there exists a q-power m=m(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=m(x)$$\end{document} such that the element xm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^m$$\end{document} is a bounded Engel element. We prove that G is locally virtually nilpotent. Further, let d, n be positive integers and w a non-commutator word. Assume that G is a d-generator residually finite group in which all w-values are n-Engel. We show that the verbal subgroup w(G) has {d,n,w}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d,n,w\}$$\end{document}-bounded nilpotency class.