Minimizing sequences of Sobolev inequalities revisited

TL;DR

This paper presents a new proof of the compactness of minimizing sequences in Sobolev inequalities at the critical level, using a simplified concentration-compactness approach that avoids complex refinements.

Contribution

It introduces a streamlined proof technique for the compactness of Sobolev inequality minimizers, simplifying existing methods.

Findings

Proves compactness of minimizing sequences in critical Sobolev inequalities

Uses a simplified concentration-compactness argument without embedding refinements

Provides a more accessible proof approach for Sobolev inequality minimizers

Abstract

We give a new proof of the compactness of minimizing sequences of the Sobolev inequalities in the critical case. Our approach relies on a simplified version of the concentration-compactness principle, which does not require any refinement of the Sobolev embedding theorem.