A profile decomposition for the limiting Sobolev embedding

TL;DR

This paper develops a general profile decomposition for bounded sequences in Sobolev spaces on compact Riemannian manifolds, extending previous results to arbitrary sequences and providing a detailed structure of their asymptotic behavior.

Contribution

It generalizes the known profile decomposition for Sobolev embeddings to all bounded sequences in $H^{1,2}(M)$, not just special cases.

Findings

Constructed a profile decomposition for arbitrary sequences in $H^{1,2}(M)$

Extended Struwe's profile decomposition to more general sequences

Provides a structured understanding of the asymptotic behavior of sequences

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 note we construct a profile decomposition for arbitrary sequences in the Sobolev space $H^{1,2}(M)$ of a compact Riemannian manifold, relative to the embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known profile decomposition of Struwe ([Proposition 2.1]{Struwe}) to the case of arbitrary bounded sequences.