Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation \begin{equation*} -\Big( u'/{ \sqrt{1-{u'}^2}}\Big)' = f(t,u). \end{equation*} Depending on the behaviour of $f=f(t,s)$ near $s=0$, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of $f$ is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.