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Liouville theorems for conformal $Q$-curvature equations

Meiqing Xu, Hui Yang

Abstract

In this paper, we study the non-existence of positive solutions for the following conformal $Q$-curvature equation \begin{equation*} (-Δ)^σu = K(x) u^{\frac{n+2σ}{n-2σ}} \quad \text{in } \mathbb{R}^n, \end{equation*} where $ σ\in (0, n/2)$ is a real number. When $σ=1$, this equation reduces to the well-known scalar curvature equation arising from the prescribed scalar curvature problem. For general $σ\in (0, n/2)$, it appears in the study of prescribing $Q$-curvature. We establish Liouville theorems under various assumptions on the $Q$-curvature $K(x)$ by developing a unified approach applicable to all $σ\in (0, n/2)$. Our method successfully addresses the challenges posed by the absence of ODE tools in the fractional regime and the lack of a classification of Delaunay-type singular solutions for the general fractional Yamabe equation.

Liouville theorems for conformal $Q$-curvature equations

Abstract

In this paper, we study the non-existence of positive solutions for the following conformal -curvature equation \begin{equation*} (-Δ)^σu = K(x) u^{\frac{n+2σ}{n-2σ}} \quad \text{in } \mathbb{R}^n, \end{equation*} where is a real number. When , this equation reduces to the well-known scalar curvature equation arising from the prescribed scalar curvature problem. For general , it appears in the study of prescribing -curvature. We establish Liouville theorems under various assumptions on the -curvature by developing a unified approach applicable to all . Our method successfully addresses the challenges posed by the absence of ODE tools in the fractional regime and the lack of a classification of Delaunay-type singular solutions for the general fractional Yamabe equation.
Paper Structure (8 sections, 23 theorems, 316 equations, 1 figure)

This paper contains 8 sections, 23 theorems, 316 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivalence of the PDE and its integral representation
  3. Liouville theorem for radially symmetric Q-curvature
  4. Liouville theorem for non-radially symmetric Q-curvature
  5. Liouville theorem for sign-changing symmetric Q-curvature
  6. Values of gm (u0,u0) in Lemma
  7. Definition and some properties of the hypergeometric function
  8. Computation of the Pohozaev integral in the fractional case

Key Result

Theorem 1.1

Let $0< \sigma <n/2$ and $u \in \mathcal{L}_\sigma (\mathbb{R}^n) \cap C(\mathbb{R}^n)$ be a nonnegative distributional solution to sobolev eq 7. Suppose that $K(x)=K(|x|) \in C^1[0, \infty)$ is nonnegative and radially nondecreasing. Then $u\equiv 0$ in $\mathbb{R}^n$ unless $K$ is a constant.

Figures (1)

  • Figure 1: $T$, $H$ and $\Sigma_\lambda$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: ao2020removability
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 35 more