Low dimensional compact embeddings of symmetric Sobolev spaces and 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 $L^\infty(\Omega)$. 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 $L^\infty(\Omega)$. 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 result are obtained in the case of partial symmetry.