Existence and symmetry of extremals for the high order Hardy-Sobolev-Maz'ya inequalities
Guozhen Lu, Chunxia Tao
TL;DR
The paper addresses the existence and symmetry of extremals for the high-order critical Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_{+}$ with $k\ge2$ and $n\ge2k+2$ by transferring the problem to a Poincaré-Sobolev inequality on hyperbolic space $\mathbb{B}^n$ via a duality framework. It develops a Lions-type concentration-compactness theory for radially decreasing sequences on hyperbolic space, combined with Helgason-Fourier analysis and the Riesz rearrangement inequality, to overcome compactness issues and establish the existence of extremals; the extremal on $\mathbb{B}^n$ then yields an extremal for the original Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_{+}$ through the established equivalence. The authors further apply these results to the high-order Brezis-Nirenberg problem on hyperbolic space involving GJMS operators $P_k$, proving the existence of positive radially symmetric solutions at the critical parameter $\alpha=\prod_{i=1}^{k}\frac{(2i-1)^2}{4}$ under the same dimensional constraints, and they extend the framework to the subcritical case $2<p<\frac{2n}{n-2k}$, where extremals exist as well. The work provides a robust hyperbolic-space strategy for sharp high-order inequalities and their extremals, with implications for symmetry and existence of solutions to related geometric PDEs on hyperbolic spaces.
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: Poincaré-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 $k\ge 2$): $$ P_{k}\left(f\right)-αf=|f|^{p-2}f $$ at the critical situation $α=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ when either $2k+2\leq n$ and $p=\frac{2n}{n-2k}$ or $2k<n$ and $2<p<\frac{2n}{n-2k}$.