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

Guozhen Lu, Chunxia Tao|arXiv (Cornell University)|Feb 5, 2026

Nonlinear Partial Differential Equations0 citations

TL;DR

The paper proves existence and symmetry of extremals for the high-order critical Hardy-Sobolev-Maz’ya inequalities on hyperbolic space, and uses a duality/concentration-compactness framework to transfer results to the upper half-space and derive symmetric Brezis-Nirenberg type solutions for GJMS operators.

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 $n\geq 2k+2$: $$\int_{\mathbb{R}^{n}_{+}}| abla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1 ight)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,\frac{2n}{n-2k}} \left(\int_{\mathbb{R}^{n}_{+}}|u|^{\frac{2n}{n-2k}}dx ight)^{\frac{n-2k}{n}}. $$ 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 $k\ge 2$): $$ P_{k}\left(f ight)-αf=|f|^{p-2}f $$ at the critical situation $α=\prod\limits_{i=1}^{k}\frac{\left(2i-1 ight)^2}{4}$ when either $2k+2\leq n$ and $p=\frac{2n}{n-2k}$ or $2k

Motivation & Objective

• Motivate the study of extremals for high-order Hardy-Sobolev-Maz’ya (HSM) inequalities on the upper half-space and its challenges.
• Develop a hyperbolic-space framework to obtain existence and symmetry of extremals for the critical HSM inequality.
• Establish a duality theory and concentration-compactness principles in the hyperbolic setting to overcome lack of compactness.
• Apply the extremal results to obtain positive symmetric solutions for high-order Brezis-Nirenberg problems involving GJMS operators.

Proposed method

• Translate the HSM inequality on the upper half-space to an equivalent Poincaré-Sobolev inequality on the hyperbolic space 01n and define the associated Poincare9-Sobolev framework.
• Develop a duality theory for minimizing sequences linking the hyperbolic Poincare9-Sobolev problem to Hardy-Littlewood-Sobolev type inequalities on 01n.
• Prove a concentration-compactness principle for radial functions in hyperbolic space and rule out vanishing and dichotomy phenomena.
• Use Helgason-Fourier analysis and Riesz rearrangement on hyperbolic space to obtain compactness and symmetry of extremals.
• Derive symmetry (radial about a point) and monotonicity of extremals from the minimizing sequence analysis.
• Apply the extremal results to the hyperbolic Brezis-Nirenberg problem for P_k (GJMS operators) at the critical parameter.

Experimental results

Research questions

• RQ1 Does the high-order critical Hardy-Sobolev-Mazya inequality on the upper half-space admit extremals for n e 2k+2 and k 1 2?
• RQ2 Can extremals be obtained via an equivalent hyperbolic Poincar-Sobolev inequality, and are they radially symmetric about some point in hyperbolic space?
• RQ3 Can a concentration-compactness framework be adapted to the hyperbolic setting to exclude vanishing and dichotomy for minimizing sequences?
• RQ4 Do the extremals yield positive symmetric solutions to the high-order Brezis-Nirenberg problem for GJMS operators P_k on 01n?

Key findings

• There exists a positive extremal function for the high-order critical Poincare9-Sobolev inequality on the hyperbolic space for n
  35 2k+2.
• The extremal on 01n is radially symmetric and monotone decreasing about some point in 01n, and this transfers to an extremal for the HSM inequality on the upper half-space.
• A positive extremal function exists for the corresponding critical HSM inequality on R^n_+ when n 5 2k+2, via the hyperbolic equivalence.
• For subcritical ranges 2<p<2n/(n-2k) with n>2k, extremals exist for both the hyperbolic Poincar-Sobolev inequality and the corresponding upper half-space inequality.
• The results yield existence of positive symmetric solutions to the Brezis-Nirenberg type equation P_k(f) - 35 f = |f|^{p-2} f on the hyperbolic space for the critical
  b1 parameter b product of (2i-1)^2/4, with specified dimensional constraints.
• These findings establish symmetry and existence in higher-order HSM problems, extending prior first-order results to k e 2.