A Gamma convergence approach to the critical Sobolev embedding in variable exponent spaces

In this paper, we study the critical Sobolev embeddings W1,p(⋅)(Ω)⊂Lp⁎(⋅)(Ω)W1,p(⋅)(Ω)⊂Lp⁎(⋅)(Ω) for variable exponent Sobolev spaces from the point of view of the Γ-convergence. More precisely we determine the Γ-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant.