Profile decomposition in Sobolev spaces of non-compact manifolds

Abstract

For many known non-compact embeddings of two Banach spaces , 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.