Asymptotics for optimizers of the fractional Hardy-Sobolev inequality

Optimizers $u$ in the Hardy-Sobolev inequality for the space $dot{W}^{s,p}(R^N)$ with order of differentiability $sin ]0,1[$ are considered. After proving the existence through concentration-compactness, the pointwise asymptotic $u(x)simeq |x|^{- rac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $uin dot{W}^{s,gamma}(R^N)$ for all $gammain ] rac{N(p-1)}{N-s}, p]$ are derived. These estimates turn out to be optimal as $s o 1^-$, in which case optimizers are explicitly known.