Asymptotic for optimizers of the fractional Hardy-Sobolev inequality

Abstract

We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space ${\dot{W}}^{s,p}({\mathbb{R}}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise asymptotic $u(x)\simeq |x{|}^{-\frac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $u\in {\dot{W}}^{s,\gamma }({\mathbb{R}}^N)$ for all $\gamma >\frac{N(p-1)}{N-s}$. These estimates are optimal in the limit $s\to 1^{-}$, in which case optimizers are explicitly known.

Extended and corrected version. Comments welcome

Cited by in corpus (1)

• The Brezis-Nirenberg problem for the fractional $p$-Laplacian involving critical Hardy-Sobolev exponents