  We prove the existence of a pair of positive radial solutions for the Neumann
boundary value problem \begin{equation*} \begin{cases} \,
\mathrm{div}\,\Biggl{(} \dfrac{\nabla u}{\sqrt{1- | \nabla u |^{2}}}\Biggr{)} +
\lambda a(|x|)u^p = 0, & \text{in $B$,} \\ \, \partial_{\nu}u=0, & \text{on
$\partial B$,} \end{cases} \end{equation*} where $B$ is a ball centered at the
origin, $a(|x|)$ is a radial sign-changing function with $\int_B
a(|x|)\,\mathrm{d}x < 0$, $p>1$ and $\lambda > 0$ is a large parameter. The
proof is based on the Leray-Schauder degree theory and extends to a larger
class of nonlinearities.
