带Hardy项和一般非线性项分数阶椭圆方程的移动平面法
The Method of Moving Planes for Fractional Order Elliptic Equations with Hardy and General Nonlinear Terms
摘要: 本文应用直接移动平面法，研究带Hardy项的分数阶拉普拉斯方程的正解的对称性和单调性。首先，关于某一点作Kelvin变换，然后建立了狭窄区域上的极值原理和无穷远处衰减原理，利用这一原理和移动平面法得到正解关于某一点对称并且关于这一点先增后减的结果。
Abstract: In this paper, we study the symmetry and monotonicity of positive solutions for fractional Laplace equations involving the Hardy potential. Firstly, the Kelvin transform is performed on a certain point, and then we establish a narrow region principle and decay at infinity principle. By using this principle and the moving plane method, the result that the positive solution is symmetric about a certain point and first increases and then decreases is obtained.
文章引用：张晓亚. 带Hardy项和一般非线性项分数阶椭圆方程的移动平面法[J]. 应用数学进展, 2023, 12(9): 3804-3813. https://doi.org/10.12677/AAM.2023.129374

参考文献

[1] Berestycki, H. and Nirenberg, L. (1991) On the Method of Moving Planes and the Sliding Method. Boletim da Sociedade Brasileira de Matemática, 22, 1-37. [Google Scholar] [CrossRef] 
[2] Chen, W., Li, C. and Li, G. (2017) Maximum Principles for a Fully Nonlinear Fractional Order Equation and Symmetry of Solutions. Calculus of Variations and Partial Differential Equations, 56, Article No. 29. [Google Scholar] [CrossRef] 
[3] Gidas, B., Ni, W. and Nirenberg, L. (1981) Symmetry of Positive Solutions of Nonlinear Elliptic Equations in Rn, Mathematical Analysis and Applications. Vol. 7a, Advances in Mathe-matics, Supplementary Studies. Academic Press, New York.
[4] Caffarelli, L. and Silvestre, L. (2007) An Extension Problem Related to the Fractional Laplacian. Communications in Partial Differential Equations, 32, 1245-1260. [Google Scholar] [CrossRef] 
[5] Chen, W. and Zhu, J. (2016) Indefinite Fractional Elliptic Prob-lem and Liouville Theorems. Journal of Differential Equations, 260, 4758-4785. [Google Scholar] [CrossRef] 
[6] Chen, W., Li, C. and Ou, B. (2005) Classification of Solutions for a System of Integral Equations. Communications in Partial Differential Equations, 30, 59-65. [Google Scholar] [CrossRef] 
[7] Ma, L. and Chen, D. (2008) Radial Symmetry and Monotonicity Results for an Integral Equation. Journal of Mathematical Analysis and Applications, 342, 943-949. [Google Scholar] [CrossRef] 
[8] Chen, W., Li, C. and Ou, B. (2005) Qualitative Properties of Solu-tions for an Integral Equation. Discrete and Continuous Dynamical Systems, 12, 347-354. [Google Scholar] [CrossRef] 
[9] Fall, M.M. (2020) Semilinear Elliptic Equations for the Fractional Laplacian with Hardy Potential. Nonlinear Analysis, 193, Article ID: 111311. [Google Scholar] [CrossRef] 
[10] Dai, W. and Qin, G. (2020) Liouville Type Theorems for Elliptic Equations with Dirichlet Conditions in Exterior Domains. Journal of Differential Equations, 269, 7231-7252. [Google Scholar] [CrossRef] 
[11] Jarohs, S. and Weth, T. (2016) Symmetry via Antisymmetric Maxi-mum Principles in Nonlocal Problems of Variable Order. Annali di Matematica, 195, 273-291. [Google Scholar] [CrossRef] 
[12] Chen, W., Li, C. and Li, Y. (2017) A Direct Method of Moving Planes for Fractional Laplacian. Advances in Mathematics, 308, 404-437. [Google Scholar] [CrossRef] 
[13] Chen, W., Fang, Y. and Yang, R. (2015) Liouville Theorems In-volving the Fractional Laplacian on a Half Space. Advances in Mathematics, 274, 167-198. [Google Scholar] [CrossRef] 
[14] Bogdan, K., Grzywny, T., Jakubowski, T. and Pilarczyk, D. (2019) Fractional Laplacian with Hardy Potential. Communications in Partial Differential Equations, 44, 20-50. [Google Scholar] [CrossRef] 
[15] Brandle, C., Colorado, E., de Pablo, A. and Sanchez, U. (2013) A Concave-Convex Elliptic Problem Involving the Fractional Laplacian. Proceedings of the Royal Society of Ed-inburgh Section A, 143, 39-71. [Google Scholar] [CrossRef] 
[16] Cai, M. and Ma, L. (2018) Moving Planes for Nonlinear Frac-tional Laplacian Equation with Negative Powers. Discrete and Continuous Dynamical Systems, 38, 4603-4615. [Google Scholar] [CrossRef] 
[17] Wang, G., Ren, X., Bai, Z. and Hou, W. (2019) Radial Symmetry of Standing Waves for Nonlinear Fractional Hardy-Schrödinger Equation. Applied Mathematics Letters, 96, 131-137. [Google Scholar] [CrossRef] 
[18] 叶方琪. 分数阶Hartree方程负解的对称性[J]. 应用数学进展, 2022, 11(3): 980-990. [Google Scholar] [CrossRef] 

为你推荐