Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity
Pierpaolo Esposito, Nassif Ghoussoub, Yujin Guo · Communications on Pure and Applied Mathematics · 2007
Abstract 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 − Δ u = λ f ( x )/(1 − u ) 2 on a bounded domain Ω ⊂ ℝ N , which models—among other things—a simple electrostatic microelectromechanical 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 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 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. © 2007 Wiley Periodicals, Inc.