On the Sobolev embedding theorem for variable exponent spaces in the   critical range
  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. 
Julian Fernandez Bonder, Nicolas Saintier, Analia Silva
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