The effect of curvature on the best constant in the Hardy–Sobolev inequalities

Abstract

We address the question of attainability of the best constant in the following Hardy–Sobolev inequality on a smooth domain Ω of $$\mathbb{R}^n$$ : $$ \mu_s(\Omega):=\text{inf}\left\{\int_\Omega |\nabla u|^2 dx; u\in H_{1,0}^2 (\Omega)\;\text{and}\;\int_{\Omega}\frac{|u|^{2^{\star}}}{|x|^s}dx=1\right\}$$ when $$0 < s < 2,\; 2^{\star}:= 2^{\ast}(s) = \frac{2(n-s)}{n-2},$$ and when 0 is on the boundary ∂Ω. This question is closely related to the geometry of ∂Ω, as we extend here the main result obtained in [GhK] by proving that at least in dimension n ≥ 4, the negativity of the mean curvature of ∂Ω at 0 is sufficient to ensure the attainability of μ s (Ω). Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions corresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [GhR2].