Asymptotic for optimizers of the fractional Hardy-Sobolev inequality

2016/09/07 by Marano, Salvatore, Mosconi, Sunra · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1609.01869

Abstract

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

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