关于分数阶椭圆方程的一类 Liouville 定理——献给邓引斌教授 70 寿辰

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (4): 1548-1553.

Previous Articles Next Articles

A Class of Liouville-Type Theorem for Fractional Elliptic Equations

Qing Ren¹(), Xian Yang²^,^*()

¹ Guangxi Technological College of Machinery and Electricity, Nanning 530007
² School of Mathematics & Center for Applied Mathematics of Guangxi (Guangxi University), Nanning 530004

Received:2026-01-04 Revised:2026-01-19 Online:2026-08-26 Published:2026-06-10
Contact: Xian Yang E-mail:renqing@gxcme.edu.cn;yangxian@gxu.edu.cn
Supported by:
NSFC(12501136)

Abstract

Abstract:

In this paper, we study the following fractional equation

($\mathcal{P}$)$ (-\Delta)^su=\chi_Hf(u)+g(u),2, \quad x\in{\mathbb{R}}^{N}, $

where $s\in[\frac{1}{2}, 1)$, $N\ge2$, $f,g\in C(\mathbb{R},\mathbb{R})$, $\chi_H$ is the characteristic function of a half space $H$ in ${\mathbb{R}}^{N}$. Under some loose and natural conditions on $f$ and $g$, we show that ($\mathcal{P}$) admits no nontrivial nonnegative weak solution in $H^s({\mathbb{R}}^{N})$. We do not require strong regularity assumptions on the solutions we study.

Key words: fractional elliptic equations, Liouville theorem, half space, weak solutions

CLC Number:

O175.2

Qing Ren, Xian Yang. A Class of Liouville-Type Theorem for Fractional Elliptic Equations[J].Acta mathematica scientia,Series A, 2026, 46(4): 1548-1553.

Trendmd

References 13

[1]	Chen W, D'Ambrosio L, Li Y. Some Liouville theorems for the fractional Laplacian. Nonlinear Anal, 2015, 121: 370-381 doi: 10.1016/j.na.2014.11.003
[2]	Chen W, Fang Y, Yang R. Liouville theorems involving the fractional Laplacian on a half space. Adv Math, 2015, 274: 167-198 doi: 10.1016/j.aim.2014.12.013
[3]	Chen W, Li Y, Ma P. The Fractional Laplacian. Hackensack, NJ: World Scientific Publishing Co Pte Ltd, 2019
[4]	Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521-573 doi: 10.1016/j.bulsci.2011.12.004
[5]	Deng Y, Peng S, Yang X. Semi-classical states for fractional Choquard equations with decaying potentials. Commun Contemp Math, 2025, 27: Art 2450046
[6]	Dipierro S, Medina M, Valdinoci E. Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$. Pisa: Edizioni della Normale, 2017
[7]	Duong A T, Nguyen V H. Liouville type theorems for some fractional elliptic problems. Nonlinear Anal, 2021, 210: Art 112383
[8]	Fall M M, Weth T. Monotonicity and nonexistence results for some fractional elliptic problems in the half-space. Commun Contemp Math, 2016, 18: Art 1550012
[9]	Fall M M, Weth T. Nonexistence results for a class of fractional elliptic boundary value problems. J Funct Anal, 2012, 263: 2205-2227
[10]	Felmer P, Quaas A, Tan J. Positive solutions of the nonlinear Schrödinger equations with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142: 1237-1262
[11]	Laskin N, Fractional quantum mechanics and Levy path integrals. Phys Lett A, 2000, 268: 298-305 doi: 10.1016/S0375-9601(00)00201-2
[12]	Quaas A, Xia A. A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian. Nonlinearity, 2016, 29: 2279-2297 doi: 10.1088/0951-7715/29/8/2279
[13]	Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60: 67-112 doi: 10.1002/cpa.v60:1

[1]	Xiaohui Yu. Weighted Pucci Operator and Nonlinear Liouville Theorems [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1814-1824.
[2]	Yibin Ren, Yuying Chen, Tian Chong. Gradient Estimates for Foliated Subelliptic Harmonic Maps [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1977-1984.
[3]	Ren Chenchen, Yang Sudan. Existence and Uniqueness of Solutions for Sub-Linear Heat Equations with Almost Periodic Coefficients [J]. Acta mathematica scientia,Series A, 2025, 45(1): 31-43.
[4]	Tong Tang,Cong Niu. Global Existence of Weak Solutions to the Quantum Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2022, 42(2): 387-400.
[5]	Kexin Luo,Shaoyong Lai. Global Weak Solutions to a High-Order Camass-Holm Type Equation [J]. Acta mathematica scientia,Series A, 2022, 42(2): 427-441.
[6]	Qinghua Zhang,Yueping Zhu. Weighted Temporal-Spatial Estimates of the Stokes Semigroup with Applications to the Non-Stationary Navier-Stokes Equation in Half-Space [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1657-1670.
[7]	Weilin Zou,Yuanchun Ren,Meipin Xiao. Regularizing Effect of L¹ Interplay Between Coefficients in Nonlinear Degenerate Elliptic Equations [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1405-1414.
[8]	Daoguo Zhou. Regularity Criteria in Lorentz Spaces for the Three Dimensional Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1396-1404.
[9]	Hongqiao Li. Liouville Type Theorem for Hartree Equations in Half Spaces [J]. Acta mathematica scientia,Series A, 2021, 41(2): 388-401.
[10]	Jiao Luo,Qian Qi,Hong Luo. Weak Solutions to Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1599-1611.
[11]	Qianqiu Wu,Lianggen Hu. Liouville Type Theorems for Stable Solutions of the Degenerate Elliptic System with Weight [J]. Acta mathematica scientia,Series A, 2020, 40(1): 156-168.
[12]	Kai Li,Han Yang,Fan Wang. Study on Weak Solution and Strong Solution of Incompressible MHD Equations with Damping in Three-Dimensional Systems [J]. Acta mathematica scientia,Series A, 2019, 39(3): 518-528.
[13]	Zhang Shengui. Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator [J]. Acta mathematica scientia,Series A, 2018, 38(3): 514-526.
[14]	Xing Chao, Ren Mengzhang, Luo Hong. The Existence of Global Weak Solutions to Thermohaline Circulation Equations [J]. Acta mathematica scientia,Series A, 2018, 38(2): 284-290.
[15]	Tang Sufang. Liouville Type Theorem for an Integral System on a Half Space [J]. Acta mathematica scientia,Series A, 2017, 37(4): 647-662.

A Class of Liouville-Type Theorem for Fractional Elliptic Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 13

Related Articles 15

Metrics

Comments

Recommended 0