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Low-dimensional compact embeddings of symmetric Sobolev spaces with applications

Published online by Cambridge University Press:  04 April 2011

Francesca Faraci
Affiliation:
Dipartimento di Matematica e Informatica, Universitá degli Studi di Catania, Viale Andrea Doria 6, 95125 Catania, Italyffaraci@dmi.unict.it
Antonio Iannizzotto
Affiliation:
Dipartimento di Matematica e Informatica, Universitá degli Studi di Catania, Viale Andrea Doria 6, 95125 Catania, Italyiannizzotto@dmi.unict.it
Alexandru Kristály
Affiliation:
Department of Economics, Babeş-Bolyai University, Str. Teodor Mihali, nr. 58–60, 400591 Cluj-Napoca, Romaniaalexandrukristaly@yahoo.com
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Abstract

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If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L(Ω). 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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2011