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We investigate a natural approximation by subcritical Sobolev embeddings ofthe Sobolev quotient for the fractional Sobolev spaces $H^s$ for any $0<s<N/2$,using $\Gamma$-convergence techniques. We show that, for such approximations,optimal functions always exist and exhibit a concentration effect of the $H^s$energy at one point.

Subcritical approximation of a Yamabe type non local equation: a Gamma-convergence approach