A threshold phenomenon for embeddings of $H^m_0$ into Orlicz spaces

TL;DR

This paper investigates the concentration-compactness behavior of critical points in the Adams-Moser-Trudinger embedding of Sobolev spaces into Orlicz spaces, revealing a threshold phenomenon related to the norms of these functions.

Contribution

It establishes a threshold phenomenon for embeddings of $H^m_0$ into Orlicz spaces, showing that non-precompact sequences have norms bounded below by a positive constant.

Findings

Non-precompact sequences have $H^m_0$-norms bounded below by a positive constant.

The study characterizes the concentration-compactness behavior in the critical embedding.

A threshold phenomenon is identified for the embedding of Sobolev spaces into Orlicz spaces.

Abstract

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.