Removable Singularities for Quasilinear Elliptic Equations with Source Terms Involving the Solution and Its Gradient

This paper establishes a removable singularity theorem for the quasilinear elliptic equations with source terms like -Δpu=a|u|q+b|∇u|s+c|u|σ|∇u|τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta _p u = a |u|^q + b|\nabla u|^s + c|u|^\sigma |\nabla u|^\tau \end{aligned}$$\end{document}with nonnegative bounded Borel measurable functions a, b, c and positive numbers q,s,σ,τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q,s,\sigma ,\tau $$\end{document}. In particular, we give upper bounds of exponents q,s,σ,τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q,s,\sigma ,\tau $$\end{document} and a sharp growth condition for nonnegative weak solutions in RN\E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N{\setminus } E$$\end{document} to be extended to the whole of 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} as solutions, when E is a compact set satisfying a uniform Minkowski condition.