Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space

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.