Rotationally symmetric solutions for shallow membrane caps

We discuss the displacements and stresses in a shallow membrane cap with either displacement or stress prescribed on the boundary and subjected to a small applied pressure. It is shown that if the prescribed boundary stress is sufficiently large there is a rotationally symmetric solution and if the applied pressure is large there is a solution to the problem with zero displacement prescribed on the boundary. However, there are situations in which there are no rotationally symmetric solutions. One case is described in which there are no rotationally symmetric solutions unless the prescribed boundary stress or prescribed boundary displacement is large.