  We study the initial value problem $$ \begin{cases}
r^{-(\gamma-1)}\left(r^{\alpha}|u'|^{\beta-1}u'\right)'=\frac{1}{f(u)} &
\textrm{for}\ 0<r<r_0,\\ u(r)>0 & \textrm{for}\ 0<r<r_0,\\ u(0)=0, \end{cases}
$$ for $\gamma>\alpha>\beta\geq 1$ and $f\in C[0,\bar u)\cap C^2(0,\bar u)$,
$f(0)=0$, $f(u)>0$ on $(0, \bar u)$ and $f$ satisfies certain assumptions which
include the standard case of pure power nonlinearities encountered in the study
of Micro-Electromechanical Systems (MEMS). We obtain the existence and
uniqueness of a solution $u^*$ to the above problem, the rate at which it
approaches the value zero at the origin and the intersection number of points
with the corresponding regular solutions $u(\,\cdot\,,a)$ (with $u(0,a)=a$) as
$a\to 0$.
  In particular, these results yield the uniqueness of a radial single point
rupture solution and other qualitative properties for MEMS models. The
bifurcation diagram is also investigated.
