Multiplicity and symmetry breaking for positive radial solutions of semilinear elliptic equations modelling MEMS on annular domains
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Date
2005-12-12
Authors
Feng, Peng
Zhou, Zhengfang
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
The use of electrostatic forces to provide actuation is a method of central importance in microelectromechanical system (MEMS) and in nano-electromechanical systems (NEMS). Here, we study the electrostatic deflection of an annular elastic membrane. We investigate the exact number of positive radial solutions and non-radially symmetric bifurcation for the model
-Δu = λ/(1-u)2 in Ω, u = 0 on ∂Ω,
where Ω = {x ∈ ℝ2 : ∊ < |x| < 1}. The exact number of positive radial solutions maybe 0, 1, or 2 depending on λ. It will be shown that the upper branch of radial solutions has non-radially symmetric bifurcation at infinitely many λN ∈ (0, λ*). The proof of the multiplicity result relies on the characterization of the shape of the time-map. The proof of the bifurcation result relies on a well-known theorem due to Kielhöfer.
Description
Keywords
Radial solution, Symmetry breaking, Multiplicity, MEMS
Citation
Feng, P., & Zhou, Z. (2005). Multiplicity and symmetry breaking for positive radial solutions of semilinear elliptic equations modelling MEMS on annular domains. <i>Electronic Journal of Differential Equations, 2005</i>(146), pp. 1-14.