A Micro electro-mechanical systems (MEMS) capacitor is a microscopic device consisting of two plates held opposite one other. The lower plate is immobile while the upper plate is fixed along its edges but free to deflect in the presence of an electric potential towards the lower plate. The deflecting upper plate may reach a stable equilibrium, however, if the applied potential exceeds a threshold, known as the pull-in voltage, the upper plate will touchdown on the lower plate. This event is crucial for the operation of certain devices ( e.g. switches ) but will compromise the utility of others (e.g. sensors). This loss of a stable equilibrium is known as the pull-in instability and has enjoyed extensive mathematical modeling.

When certain physical assumptions are applied, the deflection of the upper surface can be modeled as a fourth order PDE with a singular non-linearity whose solutions are studied. It is shown that the model captures the pull-in instability of the device and provides a prediction of the pull-in voltage. The bifurcation structure of the equilibrium equations are analyzed and shown to contain some very interesting structures. When the pull-in voltage is exceeded, it is demonstrated that the device may touchdown on multiple isolated points or on a continuous set of points which are predicted for certain domains by means of asymptotic expansions. This could potentially allow a MEMS device to perform very exotic tasks.