The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems and includes two positive parameters $$\lambda $$ and $$\varepsilon $$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $$\lambda $$ . Non-existence of steady states is also established when $$\varepsilon $$ is small enough and $$\lambda $$ is large enough (depending on $$\varepsilon $$ ).