# Profile decomposition in Sobolev spaces of non-compact manifolds

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## Abstract

For many known non-compact embeddings of two Banach spaces \(E\hookrightarrow F\), every bounded sequence in *E* has a subsequence that takes form of a *profile decomposition*—a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of *F*. In this paper we construct a profile decomposition for arbitrary sequences in the Sobolev space \(H^{1,2}(M)\) of a Riemannian manifold with bounded geometry, relative to the embedding of \(H^{1,2}(M)\) into \(L^{2^*}(M)\), generalizing the well-known profile decomposition of Struwe (Math Z 187:511–517, 1984, Proposition 2.1) to the case of general bounded sequence and a non-compact manifold.

## Keywords

Concentration compactness Profile decompositions Multiscale analysis## Mathematics Subject Classification

46E35 46B50 58J99 35B44 35A25## Notes

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