We prove the existence of multiple positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space
\begin{equation*}
\begin{cases}
-{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u, \nabla u)
& \hbox{ in } \Omega,
\\
u=0& \hbox{ on } \partial \Omega.
\end{cases}
\end{equation*}
Here $\Omega$ is a bounded regular domain in $\RR^N$
and the function $f=f(x,s,\xi)$ is
either sublinear, or superlinear, or sub-superlinear near $s=0$.
The proof combines topological and variational methods.
