  We study the branch of semi-stable and unstable solutions (i.e., those whose
Morse index is at most one) of the Dirichlet boundary value problem $-\Delta
u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$,
which models --among other things-- a simple electrostatic
Micro-Electromechanical System (MEMS) device. We extend the results of [11]
relating to the minimal branch, by obtaining compactness along unstable
branches for $1\leq N \leq 7$ on any domain $\Omega$ and for a large class of
"permittivity profiles" $f$ . We also show the remarkable fact that power-like
profiles $f(x) \simeq |x|^\alpha$ can push back the critical dimension N=7 of
this problem, by establishing compactness for the semi-stable branch on the
unit ball, also for $N\geq 8$ and as long as
$\alpha>\alpha_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . As a byproduct, we are
able to follow the second branch of the bifurcation diagram and prove the
existence of a second solution for $\lambda$ in a natural range. In all these
results, the conditions on the space-dimension and on the power of the profile
are essentially sharp.
