The moving plane method and the uniqueness of high order elliptic equation with GJMS operator

TL;DR

This work analyzes a high-order, conformally invariant PDE on $\mathbb{S}^n$ involving the GJMS operator, establishing that for $\alpha>1$ with $n\ge3$ (or $\alpha$ near 1 when $n=2m\ge4$) the only solution is $v_{\alpha}\equiv0$, and derives a new proof of Beckner's inequality. It develops a moving plane framework to prove radial symmetry about a solution’s critical point and uses Kazdan–Warner constraints to obtain a uniqueness result for $\alpha>1$, alongside a variational route to Beckner inequality. In the regime $\alpha\in(\tfrac12+\varepsilon,1)$, the paper proves compactness of the solution set via concentration-compactness and a high-order classification on $\mathbb{R}^n$, augmented by a detailed blow-up analysis. The critical limit $\alpha_i\to\tfrac12$ is analyzed, revealing the potential two-point blow-up at antipodal points and describing the asymptotic profile in terms of Green’s functions, thus mapping the full solution landscape for these conformally invariant equations.

Abstract

In this paper, we study the following high order elliptic equation involving the GJMS operator: \begin{align*} αP_{\mathbb{S}^n}v_α+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_α}. \end{align*} We establish that if $α>1$ and $n\geq3$, or if $α\in (1-ε_0, 1)$ with $n=2m\geq4$, then $v_α\equiv0$. As an application, we present a new proof of the classical Beckner inequality.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3: Maximum Principle