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,W-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,W-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.