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On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

Rongxun He, Wei Ke

TL;DR

The paper establishes a Hessian Hardy–Sobolev inequality for the $k$-Hessian operator $S_k(D^2u)$ on strictly $(k-1)$-convex domains, with a weighted critical exponent $k^*(s)$ that depends on the dimension and the weight parameter $s$. Using a descent gradient-flow approach, it reduces the inequality to radial profiles and develops a robust variational framework for the Dirichlet problem $S_k(D^2u)=|x|^{2sk}f(x,u)$, covering both sublinear and superlinear nonlinearities. It proves existence and regularity results for nontrivial admissible solutions in a weighted Hessian setting, builds uniform a priori estimates, and derives nonexistence results in the degenerate/subcritical regime, thereby extending the variational Hessian theory beyond the classical Hessian equation. Overall, the work blends parabolic Hessian theory, variational methods, and sharp weighted inequalities to advance Hessian-type Hardy–Sobolev theory with concrete extremals and regularity results.

Abstract

In this paper, we first prove the Hardy-Sobolev inequality for the Hessian integral by means of a descent gradient flow of certain Hessian functionals. As an application, we study the existence and regularity results of solutions to related variational problems. Our results extend the variational theory of the Hessian equation in \cite{CW01variational}.

On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

TL;DR

The paper establishes a Hessian Hardy–Sobolev inequality for the -Hessian operator on strictly -convex domains, with a weighted critical exponent that depends on the dimension and the weight parameter . Using a descent gradient-flow approach, it reduces the inequality to radial profiles and develops a robust variational framework for the Dirichlet problem , covering both sublinear and superlinear nonlinearities. It proves existence and regularity results for nontrivial admissible solutions in a weighted Hessian setting, builds uniform a priori estimates, and derives nonexistence results in the degenerate/subcritical regime, thereby extending the variational Hessian theory beyond the classical Hessian equation. Overall, the work blends parabolic Hessian theory, variational methods, and sharp weighted inequalities to advance Hessian-type Hardy–Sobolev theory with concrete extremals and regularity results.

Abstract

In this paper, we first prove the Hardy-Sobolev inequality for the Hessian integral by means of a descent gradient flow of certain Hessian functionals. As an application, we study the existence and regularity results of solutions to related variational problems. Our results extend the variational theory of the Hessian equation in \cite{CW01variational}.
Paper Structure (6 sections, 12 theorems, 153 equations)

This paper contains 6 sections, 12 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hessian Hardy-Sobolev Inequality
  4. The Sublinear Case
  5. The Superlinear Case
  6. Nonexistence Results

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$ be any smooth $(k-1)$-convex domain containing the origin. Suppose that $n>2k$, $-1\leqslant s\leqslant0$ and $k^*=k^*(s)>0$ be such that Then it holds for all $u\in\Phi_0^k(\Omega)$, where the constant $C$ depends only on $n,k$ and $s$. In particular, if $-1<s\leqslant0$, the best constant can be attained when $\Omega=\mathbb{R}^n$ by the function with some posi

Theorems & Definitions (16)

  • Theorem 1.1: Hessian Hardy-Sobolev inequality
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 6 more