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Liouville type theorems involving fractional order systems.

Authors :
Liao, Qiuping
Liu, Zhao
Wang, Xinyue
Source :
Advanced Nonlinear Studies; Apr2024, Vol. 24 Issue 2, p399-414, 16p
Publication Year :
2024

Abstract

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (− Δ) α / 2 u (x) = f (u (x) , v (x)) , x ∈ R n , (− Δ) α / 2 v (x) = g (u (x) , v (x)) , x ∈ R n. Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang ("A direct method of moving spheres on fractional order equations," J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu ("Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space," Potential Anal., vol. 46, pp. 569–588, 2017). [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
15361365
Volume :
24
Issue :
2
Database :
Complementary Index
Journal :
Advanced Nonlinear Studies
Publication Type :
Academic Journal
Accession number :
176696908
Full Text :
https://doi.org/10.1515/ans-2023-0108