A Profile Decomposition for the Limiting Sobolev Embedding

For many known non-compact embeddings of two Banach spaces E ,! F, every bounded sequence in E has a subsequence that takes the 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 note we construct a profile decomposition for arbitrary sequences in the Sobolev space H1;2(M) of a compact Riemannian manifold, relative to the embedding of H1;2(M) into L^2 (M), generalizing the well-known profile decomposition of Struwe [12, Proposition 2.1] to the case of arbitrary bounded sequences.