On the Sobolev embedding theorem for variable exponent spaces in the critical range

TL;DR

This paper investigates the Sobolev embedding theorem in variable exponent spaces at critical exponents, establishing conditions for extremal existence using advanced concentration-compactness techniques.

Contribution

It provides new conditions on the best constant ensuring extremals in variable exponent Sobolev embeddings at critical ranges.

Findings

Derived conditions for extremal existence.

Refined concentration-compactness estimates.

Adapted convexity arguments for variable exponents.

Abstract

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.