MULTIPLICITY AND SYMMETRY BREAKING FOR POSITIVE RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS MODELLING MEMS ON ANNULAR DOMAINS

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 -Delta u = lambda/(1 -u)(2) in Omega, u = 0 on partial derivative Omega, where Omega = {x is an element of R-2 : epsilon < vertical bar x vertical bar < 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) is an element of (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.