Compactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity

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.