  We study the effect of the parameter $\lambda$, the dimension $N$, the
profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the
question of uniqueness of the solutions to the following elliptic boundary
value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -\Delta u=
\frac{\lambda f(x)}{(1-u)^2} & \hbox{in}\Omega 0<u<1 &\hbox{in}\Omega u=0
&\hbox{on}\partial \Omega. {array}. 130pt (S)_{\lambda, f} $$ This equation has
been proposed as a model for a simple electrostatic Micro-Electromechanical
System (MEMS) device consisting of a thin dielectric elastic membrane with
boundary supported at 0 below a rigid ground plate located at height z = 1.
