We prove growth rate estimates and existence of solutions to Dirichlet problems for prescribed mean curvature equation on unbounded domains inside the complement of a cone or a parabola like region in Rn (n 2). The existence results are proved using a modified Perron’s method by which a subsolution is a solution to the minimal surface equation, while the role played by a supersolution is replaced by estimates on the uniform C0 bounds on the liftings of subfunctions on compact sets.