  We investigate the existence of positive solutions for a class of
Minkowski-curvature equations with indefinite weight and nonlinear term having
superlinear growth at zero and super-exponential growth at infinity. As an
example, for the equation \begin{equation*} \Biggl{(}
\dfrac{u'}{\sqrt{1-(u')^{2}}}\Biggr{)}' + a(t) \bigl{(}e^{u^{p}}-1\bigr{)} = 0,
\end{equation*} where $p > 1$ and $a(t)$ is a sign-changing function satisfying
the mean-value condition $\int_{0}^{T} a(t)\,\mathrm{d}t < 0$, we prove the
existence of a positive solution for both periodic and Neumann boundary
conditions. The proof relies on a topological degree technique.
