Minimization problem on the Hardy–Sobolev inequality

Abstract

We study minimization problems on Hardy–Sobolev type inequality. We consider the case where singularity is in interior of bounded domain \(\Omega \subset \mathbb {R}^N\). The attainability of best constants for Hardy–Sobolev type inequalities with boundary singularities have been studied so far, for example Ghoussoub and Kang (Ann Inst Henri Poincare Anal Non Lineaire 21(6):767–793, 2004), Ghoussoub and Robert (IMRP 21867:1–85, 2006), Ghoussoub and Robert (Trans Am Math Soc 361(9):4843–4870, 2009) etc.... According to their results, the mean curvature of \(\partial \Omega \) at singularity affects the attainability of the best constants. In contrast with boundary singularity case, in interior singularity case it is well known that the best Hardy–Sobolev constant $$\begin{aligned} \mu _s(\Omega ):=\left\{ \int _\Omega |\nabla u|^2 dx \Bigg | u \in H_0^1(\Omega ),\ \int _\Omega \frac{|u|^{2^*(s)}}{|x|^s}dx = 1 \right\} \end{aligned}$$is never achieved for all bounded domain \(\Omega \). We can see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding \(H^1(\Omega ) \hookrightarrow L^{2^*(s)}(\Omega ,|x|^{-s}dx)\) for bounded domain \(\Omega \) with \(0 \in \Omega \). In this problem, scaling invariance doesn’t hold and we can not obtain information of singularity like mean curvature.