We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem $-\Delta u = λf (x)/(1 − u)^2$ on a bounded domain $\Omega ⊂ R^N$ , which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of  relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain \Omega and for a large class of “permittivity profiles” f . We also show the remarkable fact that powerlike profiles f (x) ≃ |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N≥8 and as long as $α>α_N =3N−14−4√6/4+2√6$. 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.
Esposito, P., Ghoussoub, N., Guo, Y. (2007). Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 60(12), 1731-1768 [10.1002/cpa.20189].