Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 146, pp. 1-14.

Title: Multiplicity and symmetry breaking for positive radial 
       solutions of semilinear elliptic equations modelling 
       MEMS on annular domains
       
Authors: Peng Feng (Michigan State Univ., East Lansing, MI, USA)
         Zhengfang Zhou (Michigan State Univ., East Lansing, MI, USA)

Abstract: 
 The use of electrostatic forces to provide actuation is a method
 of central importance in microelectromechanical system (MEMS)
 and in nanoelectromechanical  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
 $$
 -\Delta u=\frac{\lambda}{(1-u)^2}\quad\hbox{in }\Omega,
 \quad  u=0 \quad\hbox{on }\partial \Omega,
 $$
 where
 $\Omega=\{x\in \mathbb{R}^2: \epsilon<|x|<1\}$.
 The exact number of positive radial solutions maybe 0, 1, or 2
 depending on $\lambda$.
 It will be shown that the upper branch of radial solutions
 has non-radially symmetric bifurcation at infinitely many
 $\lambda_N\in (0,\lambda^*)$. 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 Kielhofer.
 
Submitted August 10, 2005. Published December 12, 2005.
Math Subject Classifications: 35J65, 35J60, 35B32.
Key Words: Radial solution; symmetry breaking; multiplicity; MEMS.