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LetΛ  p    (  Ω  )   =  inf∇  u    p   p   :  u  ∈   W   0    1  ,  p(  Ω  )anduL  ∞    (  Ω  )=  1    , where Ω is a bounded domain of    R  N   ,    N  ≥  2   , and    p  >  N   . We prove thatlim   p  →  ∞Λ  p(  Ω  )    1  p    =    ρ    ∞    −  1, where ρ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions ofΛ  p    (  Ω  )converge (as    p  →  ∞   ) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured domainΩ  ∖x  *, for somex  *   ∈  Ω   .

Asymptotics for the best Sobolev constants and their extremal functions