Global Structure Of Positive Radial Solutions Of Elliptic Problem With Singular Nonlinearity

The use of electrostatic force to provide actuation is a method of central importance in microelectromechanical system (MEMS) and nanoelectiomechanical systems (NEMS), the study of electrostatic actuation has led to some interesting experimental, theoretical and numerical results. We first study the exact number and non-radially symmetric bifurcation of an elliptic problem on annular domains in 2-dimension. The exact number of positive radial solutions maybe 0, 1, 2 depending on the parameter. Moreover, it will be shown that the upper branch of radial solutions has non-radially symmetric bifurcation points. The proof of this section relies on the characterization of the shape of the time-map. In the second section, we study the structure of an elliptic problem with singular nonlinearity on the unit ball in N-dimension. We show that the maximal solution is the only positive radial solution when the parameter is small. Moreover, we can obtain the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.