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Research articles

ScienceAsia 49 (2023):ID 710-716 |doi: 10.2306/scienceasia1513-1874.2023.049

Maximum principles and Liouville theorems for fractional Kirchhoff equations

Pengyan Wanga,*, Miaomiao Caib

ABSTRACT:     In this paper, we consider the following nonlinear fractional Kirchhoff equation a + b Z Rn |(??) s 2 u| 2 dx (??) s u(x) = f (u(x)), where 0 < s < 1, a > 0 and b ? 0. We first establish a maximum principle for anti-asymmetric functions on any half space, and then obtain a Liouville theorem to the above nonlinear fractional Kirchhoff equations in the whole space. In particular, we derive key ingredients for proving the symmetry and monotonicity of positive solutions to the nonlinear fractional Kirchhoff equations, which indicate that fractional Kirchhoff De Giorgi conjecture is valid under some conditions. We believe that the results obtained here can be conveniently applied to study a variety of properties for solutions to fractional Kirchhoff equations.

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a School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000 China
b School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046 China

* Corresponding author, E-mail: wangpy@xynu.edu.cn

Received 1 Jul 2022, Accepted 5 Apr 2023