A canonical model arising in MEMS and modeling the electrostatic deflections of an elastic membrane with spatially varying dielectric properties is considered. Symmetry methods are used to determine the form of the spatially varying dielectric properties that allow for invariance under a one-parameter Lie group of transformations. In these cases it is shown how symmetry may be used to analyze the boundary value problem and construct a bifurcation diagram for the system. This bifurcation diagram relates back to the MEMS model; a non-dimensional version of the applied voltage is plotted versus the maximum deflection of the membrane.

We show among other things, that for small voltage, the stable solution of the basic nonlinear eigenvalue problem modelling a simple electrostatic MEMS is actually the unique solution, provided the domain is star-shaped and the dimension is larger or equal than 3. In two dimensions, we need the domain to be either strictly convex or symmetric. The case of a power permittivity profile is also considered. Our results, which use an approach developed by Schaaf, extend and simplify recent results by Guo and Wei.