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.

Low dimensional compact embeddings of symmetric Sobolev spaces and applications

FARACI, FRANCESCA;
2011

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11769/791
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