Radial symmetry of positive solutions in nonlinear polyharmonic Dirichlet problems

We extend the symmetry result of Gidas-Ni-Nirenberg to semilinear polyharmonic Dirichlet problems in the unit ball. In the proof we develop a new variant of the method of moving planes relying on fine estimates for the Green function of the polyharmonic operator. We also consider minimizers for subcritical higher order Sobolev embeddings. For embeddings into weighted spaces with a radially symmetric weight function, we show that the minimizers are at least axially symmetric. This result is sharp since we exhibit examples of minimizers which do not have full radial symmetry.