Liouville-type theorems for a quasilinear elliptic equation of the Hénon-type

We consider the Hénon-type quasilinear elliptic equation -Δmu=|x|aup\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${-\Delta_m u=|x|^a u^p}$$\end{document} where Δmu=div(|∇u|m-2∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_m u={\rm div}(|\nabla u|^{m-2} \nabla u)}$$\end{document}, m > 1, p > m − 1 and a≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a\geq 0}$$\end{document}. We are concerned with the Liouville property, i.e. the nonexistence of positive solutions in the whole space 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}. We prove the optimal Liouville-type theorem for dimension N < m + 1 and give partial results for higher dimensions.