Symmetric solutions to supercritical elliptic problems

When searching for solutions to Sobolev-supercritical elliptic problems, a major difficulty is the lack of Sobolev embeddings, that entrains a lack of compactness. In this talk, I will discuss how symmetry and monotonicity properties can help to overcome this obstacle. In particular, I will present a recent result concerning the existence of axially symmetric solutions to a semilinear equation, in collaboration with A. Boscaggin, F. Colasuonno and T. Weth.