  A variational approach is employed to find stationary solutions to a free
boundary problem modeling an idealized electrostatically actuated MEMS device
made of an elastic plate coated with a thin dielectric film and suspended above
a rigid ground plate. The model couples a non-local fourth-order equation for
the elastic plate deflection to the harmonic electrostatic potential in the
free domain between the elastic and the ground plate. The corresponding energy
is non-coercive reflecting an inherent singularity related to a possible
touchdown of the elastic plate. Stationary solutions are constructed using a
constrained minimization problem. A by-product is the existence of at least two
stationary solutions for some values of the applied voltage.
