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 −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ R , 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 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f . We also show the remarkable fact that power-like profiles f(x) ' |x|α 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 ≥ 8 and as long as α > αN = 3N−14−4√6 4+2 √ 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 λ in a natural range. In all these results, the conditions on the space-dimension and on the power of the profile are essentially sharp.