  We analyze the nonlinear elliptic problem $\Delta u=\frac{\lambda
f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $\R^N$ with Dirichlet boundary
conditions. This equation models a simple electrostatic Micro-Electromechanical
System (MEMS) device consisting of a thin dielectric elastic membrane with
boundary supported at 0 above a rigid ground plate located at -1. When a
voltage --represented here by $\lambda$-- is applied, the membrane deflects
towards the ground plate and a snap-through may occur when it exceeds a certain
critical value $\lambda^*$ (pull-in voltage). This creates a so-called "pull-in
instability" which greatly affects the design of many devices. The mathematical
model lends to a nonlinear parabolic problem for the dynamic deflection of the
elastic membrane which will be considered in forthcoming papers \cite{GG2} and
\cite{GG3}. For now, we focus on the stationary equation where the challenge is
to estimate $\lambda^*$ in terms of material properties of the membrane, which
can be fabricated with a spatially varying dielectric permittivity profile $f$.
Applying analytical and numerical techniques, the existence of $\lambda^*$ is
established together with rigorous bounds. We show the existence of at least
one steady-state when $\lambda < \lambda^*$ (and when $\lambda=\lambda^*$ in
dimension $N< 8$) while none is possible for $\lambda>\lambda^*$. More refined
properties of steady states --such as regularity, stability, uniqueness,
multiplicity, energy estimates and comparison results-- are shown to depend on
the dimension of the ambient space and on the permittivity profile.
