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.