On solutions of the singular minimal surface equation

Results of Bernstein type are proven for supersolutions of the singular minimal surface equation when α<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <0$$\end{document}. In particular the non-existence of “entire” minimal graphs in hyperbolic space is shown. In addition we construct a foliation of Rn×R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n\times \mathbb {R}^+$$\end{document} consisting of minimizing surfaces, and solve a Dirichlet problem for the singular minimal surface equation.