Existence and symmetry of extremals for the high order Hardy-Sobolev-Maz'ya inequalities

Abstract

In this article, we establish the existence of an extremal function for the k-th order critical Hardy-Sobolev-Maz'ya (HSM) inequalities on the upper half space $\mathbb{R}^{n+1}_{+}$ when $k\ge 2$ and $n\geq 2k+2$: $$\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,\frac{2n}{n-2k}} \left(\int_{\mathbb{R}^{n}_{+}}|u|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}. $$ The analysis of this extremal problem is challenging due to the presence of the higher order derivatives, the lack of translation invariance, the inapplicability of rearrangement techniques on the upper half-space, and the presence of a Hardy singularity along the boundary. To overcome these difficulties, instead of directly considering the HSM inequality on the upper half space, we establish the existence of an extremal for its equivalent version: Poincar\'e-Sobolev inequality on the hyperbolic space. We develop a novel duality theory of the minimizing sequences, the concentration-compactness principle for radial functions in the hyperbolic setting, which combines with the Helgason-Fourier analysis and the Riesz rearrangement inequality on the hyperbolic space, to resolve the lack of compactness issue. As an application, we also obtain the existence of positive symmetric solutions for the high order Brezis-Nirenberg equation on the entire hyperbolic space associated with the GJMS operators $P_k$ (i.e., when $k\ge 2$): $$ P_{k}\left(f\right)-\alpha f=|f|^{p-2}f $$ at the critical situation $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ when either $2k+2\leq n$ and $p=\frac{2n}{n-2k}$ or $2k<n$ and $2<p<\frac{2n}{n-2k}$.