On a fourth order elliptic problem with a singular nonlinearity

We study a nonlinear eigenvalue problem for the biharmonic operator, subject to Dirichlet boundary conditions, where the nonlinearity exhibits a singularity. We prove in the ball the existence of an extremal value of the parameter (pull-in voltage) below which there exists a minimal (classical) solution. In the extremal case we prove the existence of a finite energy solution which is unique even in a very weak sense. Above this critical value there are no solutions of any kind; a characterization of the extremal solution is then derived. Estimates on the pull-in voltage, stability properties of solutions and nonexistence results in the whole space are also established.