  Motivated by certain mathematical models for Micro-Electro-Mechanical Systems
(MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions
of nonlinear eigenvalue problems of the form $-\Delta u = \lambda f(x) F(u)$ on
a smooth bounded domain $ \Omega$ in $\IR^N$. We are mainly interested in the
{\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution
$u^*$ and how it depends on the geometry of the domain, the dimension of the
space, and the so-called {\it permittivity profile} $f$. In particular, our
results provide mathematical proofs for various observed phenomena, as well as
rigorous derivations for several estimates obtained numerically by Pelesko
\cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS
non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles
$f(x)=|x|^\alpha$.
