  Micro-Electro Mechanical Systems (MEMS) are defined as very small structures
that combine electrical and mechanical components on a common substrate. Here,
the electrostatic-elastic case is considered, where an elastic membrane is
allowed to deflect above a ground plate under the action of an electric
potential, whose strength is proportional to a parameter $\lambda$. Such
devices are commonly described by a parabolic partial differential equation
that contains a singular nonlinear source term. The singularity in that term
corresponds to the so-called "touchdown" phenomenon, where the membrane
establishes contact with the ground plate. Touchdown is known to imply the
non-existence of steady state solutions and blow-up of solutions in finite
time. We study a recently proposed extension of that canonical model, where
such singularities are avoided due to the introduction of a regularizing term
involving a small "regularization" parameter $\varepsilon$. Methods from
dynamical systems and geometric singular perturbation theory, in particular the
desingularization technique known as "blow-up", allow for a precise description
of steady-state solutions of the regularized model, as well as for a detailed
resolution of the resulting bifurcation diagram. The interplay between the two
main model parameters $\varepsilon$ and $\lambda$ is emphasized; in particular,
the focus is on the singular limit as both parameters tend to zero.
