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

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dc.creatorEsposito, Pierpaolo
dc.creatorGhoussoub, Nassif
dc.creatorGuo, Yujin
dc.date2005-11-28
dc.date.accessioned2026-07-07T06:51:50Z
dc.date.available2026-07-07T06:51:50Z
dc.descriptionWe 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 $-Δu=\frac{λf(x)}{(1-u)^2}$ on a bounded domain $Ω\subset \R^N$, 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 $1\leq N \leq 7$ on any domain $Ω$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^α$ 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 $N\geq 8$ and as long as $α>α_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . 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.
dc.description29 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/
dc.identifierhttps://arxiv.org/abs/math/0511690
dc.identifierhttp://arxiv.org/abs/math/0511690
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/105119
dc.subjectAnalysis of PDEs
dc.subject35J60, 35B40, 35J20
dc.titleCompactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity
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