In this paper, we study positive solutions of the quasilinear elliptic equationQp,A,V′[u]≜−divA(x,∇u)+V(x)|u|p−2u=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{array}{@{}rcl@{}} Q^{\\prime}_{p,\\mathcal{A},V}[u]\ riangleq -\ ext{div} {\\mathcal{A}(x,\ abla u)}+V(x)|u|^{p-2}u=0, \\end{array} $$\\end{document} in a domain Ω⊆ℝn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Omega }\\subseteq \\mathbb {R}^{n}$\\end{document}, where n ≥ 2, 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p <\\infty $\\end{document}, the divergence of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {A}$\\end{document} is the well known A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {A}$\\end{document}-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential V belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator Qp,A,V′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q^{\\prime }_{p,\\mathcal {A},V}$\\end{document}. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of Qp,A,V′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q^{\\prime }_{p,\\mathcal {A},V}$\\end{document} in a domain ω⋐Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\omega \\Subset {\\Omega }$\\end{document}, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation Qp,A,V′[u]=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q^{\\prime }_{p,\\mathcal {A},V}[u]=0$\\end{document} of minimal growth at infinity in Ω, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.