Low-dimensional compact embeddings of symmetric Sobolev spaces with applications

If Omega is an unbounded domain in R^N and p > N, the Sobolev space W^(1,p)(Omega) is not compactly embedded into L^infinity(Omega). Nevertheless, we prove that if Omega is a strip-like domain, then the subspace of W^(1,p)(Omega) consisting of the cylindrically symmetric functions is compactly embedded into L^infinity(Omega). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.