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
( u' / sqrt{1-(u')^2} )' + a(t) (e^{u^p}-1) = 0,
where p > 1 and a(t) is a sign-changing function satisfying the mean-value condition int_0^T a(t) dt < 0, we prove the existence of a positive solution for both periodic and Neumann boundary conditions. The proof relies on a topological degree technique.
