Results of Bernstein type are proven for supersolutions of the singular minimal surface equation when $$\alpha <0$$ α < 0 . In particular the non-existence of “entire” minimal graphs in hyperbolic space is shown. In addition we construct a foliation of $$\mathbb {R}^n\times \mathbb {R}^+$$ R n × R + consisting of minimizing surfaces, and solve a Dirichlet problem for the singular minimal surface equation.