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In this article, we consider the Dirichlet problem for the prescribed mean curvature equation in the Minkowski space, $$\displaylines{ -\hbox{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) =\lambda f(u) \quad \text{in } B_R,\cr u=0 \quad \text{on } \partial B_R, }$$ where $B_R:=\{x\in \mathbb{R}^N: |x|< R\}$, $\lambda>0$ is a parameter and $f:[0, \infty)\to\mathbb{R}$ is continuous. We apply some standard variational techniques to show how changes in the sign of f lead to multiple positive solutions of the above problem for sufficiently large $\lambda$.