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Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity
Article first published online: 30 APR 2007
DOI: 10.1002/cpa.20189
Copyright © 2007 Wiley Periodicals, Inc.
Issue
1097-0312/asset/cover.gif?v=1&s=27e2b2132052ff78aa82eddfab7608281350adb9 "Communications on Pure and Applied Mathematics")
Communications on Pure and Applied Mathematics
Volume 60, Issue 12, pages 1731–1768, December 2007
Additional Information
How to Cite
Esposito, P., Ghoussoub, N. and Guo, Y. (2007), Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Math., 60: 1731–1768. doi: 10.1002/cpa.20189
Publication History
- Issue published online: 24 SEP 2007
- Article first published online: 30 APR 2007
- Manuscript Received: NOV 2005
Funded by
- Ministero per l'Uni-versità e per la Ricerca Scientifica e Tecnologica (MURST) Project “Variational Methods and Nonlinear Differential Equations”
- Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship
- Natural Science and Engineering Research Council of Canada
- Natural Science Foundation of P.R. China. Grant Number: 10171036
- University of British Columbia Graduate Fellowship
- Abstract
- References
- Cited By
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 − Δu = λf(x)/(1 − u)2 on a bounded domain Ω ⊂ ℝN, 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.