Multiplicity and symmetry breaking for positive radial solutions of semilinear elliptic equations modelling MEMS on annular domains

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 $$ -Delta u=frac{lambda}{(1-u)^2}quadhbox{in }Omega, quad u=0 quadhbox{on }partial Omega, $$ where $Omega={xin mathbb{R}^2: epsilon<|x|<1}$. 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 $lambda_Nin (0,lambda^*)$. 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.

Keywords

• Radial solution
• symmetry breaking
• multiplicity
• MEMS.