Summary: We produce a detailed proof of a result of C. V. Coffman and W. K. Ziemer [SIAM J. Math. Anal. 22, No. 4, 982–990 (1991; Zbl 0741.35010)] on the existence of positive solutions of the Dirichlet problem for the prescribed mean curvature equation \[ -\operatorname{div}(\nabla u/\sqrt{1+|\nabla u|^2)}=\lambda f(x,u)\,\,\text{in}\,\,\Omega, \qquad u=0\,\,\text{on}\,\,\partial\Omega, \] assuming that \(f\) has a superlinear behaviour at \(u=0\).