The onset of multi-valued solutions of a prescribed mean curvature equation with singular nonlinearity

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of micro-electromechanical systems (MEMS). It is observed that the bifurcation curve of solutions terminates at single deadend point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution’s gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point, however, all extra solutions are found to be multivalued. This analysis therefore suggests the dead-end is a bifurcation point associated with the onset of multivalued solutions for the system.