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On elliptic equations with N-independent stable operators

Lele Du, Minbo Yang

TL;DR

The paper studies Liouville-type results for the nonlocal elliptic equation $\sum_{i=1}^{N}(-\partial_{ii})^{s}u=u^{p}$ with $N$ independent $2s$-stable operators, establishing sharp nonexistence thresholds and symmetry properties across the whole space, half-space, and unit ball. Central to the approach are the anisotropic operator $\mathcal{I}$, its Fourier symbol $\widehat{\mathcal{I}}(\xi)=|\xi|_{2s}^{2s}$, and the Green kernel $G_s$; the authors combine scaled test-function arguments, a tailored maximum principle along coordinate directions, the integral representation $u=u^{p}*G_s$, and the moving-plane method adapted to anisotropy to derive symmetry and nonexistence results. In particular, they obtain nonexistence of positive supersolutions for $1<p\le \frac{N}{N-2s}$ in $\mathbb{R}^{N}$ and symmetry for $p>\frac{N}{N-2s}$, extend nonexistence to the half-space with threshold $\frac{N+s}{N-s}$, and prove symmetry in the unit ball, highlighting the role of decay at infinity in the absence of a Kelvin transform and the impact of non-rotational invariance on symmetry. These results advance understanding of Liouville-type phenomena for nonlocal, directionally decomposed stable operators and illuminate how anisotropy shapes solution symmetry and decay requirements.

Abstract

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space $\R^{N}$, we establish the nonexistence of positive supersolutions for $1<p\leq\frac{N}{N-2s}$. Furthermore, the symmetry of positive solutions is obtained when $p>\frac{N}{N-2s}$. It is crucial for these solutions to exhibit suitable decay at infinity to compensate for the absence of the Kelvin transform. Notably, while these solutions are symmetric, they are not radially symmetric due to the non-rotational invariance of the operator involved. Next, in the half space $\R_{+}^{N}$, we observe the nonexistence of positive supersolutions in the region $1<p\leq\frac{N+s}{N-s}$. Additionally, we find that positive solutions with appropriate decay for the Dirichlet boundary problem do not exist. Finally, we present the symmetry of positive solutions in the unit ball $B_{1}$.

On elliptic equations with N-independent stable operators

TL;DR

The paper studies Liouville-type results for the nonlocal elliptic equation with independent -stable operators, establishing sharp nonexistence thresholds and symmetry properties across the whole space, half-space, and unit ball. Central to the approach are the anisotropic operator , its Fourier symbol , and the Green kernel ; the authors combine scaled test-function arguments, a tailored maximum principle along coordinate directions, the integral representation , and the moving-plane method adapted to anisotropy to derive symmetry and nonexistence results. In particular, they obtain nonexistence of positive supersolutions for in and symmetry for , extend nonexistence to the half-space with threshold , and prove symmetry in the unit ball, highlighting the role of decay at infinity in the absence of a Kelvin transform and the impact of non-rotational invariance on symmetry. These results advance understanding of Liouville-type phenomena for nonlocal, directionally decomposed stable operators and illuminate how anisotropy shapes solution symmetry and decay requirements.

Abstract

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric -stable operators. Firstly, in the whole space , we establish the nonexistence of positive supersolutions for . Furthermore, the symmetry of positive solutions is obtained when . It is crucial for these solutions to exhibit suitable decay at infinity to compensate for the absence of the Kelvin transform. Notably, while these solutions are symmetric, they are not radially symmetric due to the non-rotational invariance of the operator involved. Next, in the half space , we observe the nonexistence of positive supersolutions in the region . Additionally, we find that positive solutions with appropriate decay for the Dirichlet boundary problem do not exist. Finally, we present the symmetry of positive solutions in the unit ball .
Paper Structure (27 sections, 24 theorems, 335 equations, 4 figures)

This paper contains 27 sections, 24 theorems, 335 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem in the whole space
  3. Problem in the half space
  4. Problem in the unit ball
  5. Preliminary
  6. Basic properties
  7. Maximum principle
  8. Integration by parts formula
  9. Fourier symbol
  10. Symmetric potential
  11. Equivalent integral equation
  12. Nonexistence and symmetry in the whole space
  13. Nonexistence of positive supersolutions: proof of Theorem \ref{['T1-1']} and \ref{['T1-2']}
  14. Proof of Theorem \ref{['T1-1']}
  15. Proof of Theorem \ref{['T1-2']}
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $0<s<1$ and $1<p\leq\frac{N}{N-2s}$. Assume $u\in C^{2}\left(\mathbb{R}^{N}\right)\cap\overline{\mathcal{L}_{s}}$ is a nonnegative solution of E1-5, then $u\equiv0$.

Figures (4)

  • Figure 1: Symmetry hyperplanes and decreasing directions when $s=\frac{1}{2}$.
  • Figure 2: Diffusion along the coordinate axis directions
  • Figure 3: The graph of $\widetilde{u}$.
  • Figure 4: $\varphi\left(x_{\infty}\pm te_{N}\right)$ does not always be $0$ after moving up the ball.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 30 more