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

2005/11/28 by Esposito, Pierpaolo, Ghoussoub, Nassif, Guo, Yujin
#35B40 #35J20 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0511690

Abstract

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 -Δu=(λf(x))/((1-u)²) on a bounded domain Ω⊂ \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≤ 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>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.

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