Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$-\Delta_{m}(u) = f(u)$$ \end{document} in Ω with zero Dirichlet boundary conditions. We prove local summability properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{1}{\mid Du \mid}$$ \end{document} and we exploit these results to give geometric characterizations of the critical set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Z = \{x \in \Omega | Du(x) = 0\}$$ \end{document}. We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0.