Uniqueness of solutions for an elliptic equation modeling MEMS

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 [13], extend and simplify recent results by Guo and Wei [7], [8].