Characterization of the lack of compactness of $H^2_{rad}(\R^4)$ into the Orlicz space

TL;DR

This paper characterizes the reasons for the failure of compactness of the radially symmetric Sobolev space $H^2_{rad}( ^4)$ when embedded into an Orlicz space, extending previous work on lower-order Sobolev spaces.

Contribution

It provides a detailed description of the lack of compactness for $H^2_{rad}( ^4)$ into Orlicz spaces, using an approach inspired by earlier results for $H^1_{rad}( ^2)$.

Findings

Identifies the specific mechanisms causing non-compactness.

Extends known results from lower-order Sobolev spaces.

Provides a framework for analyzing similar embeddings.

Abstract

This paper is devoted to the description of the lack of compactness of the Sobolev space $H^2_{rad}(\R^4)$ in the Orlicz space $\mathcal{L}(\R^4)$. The approach that we adopt to establish this characterization is in the spirit of the one adopted in the case of $H^1_{rad}(\R^2)$ into the Orlicz space $\mathcal{L}(\R^2)$ in \cite{Bahouri}.