References
- Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Transcendental Functions. vol. II. New York: McGraw-Hill; 1953.
- Fall MM, Felli V. Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential. J Funct Anal. 2014;267:1851–1877.
- Fall MM, Felli V. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Disc Contin Dyn Syst A. 2015;35:5827–5867.
- Bueno H, Medeiros A, Pereira G. Pohozaev-type identities for a pseudo-relativistic Schördinger operator and applications. Complex Var Elliptic Equ. 2022;67:2481–2506.
- Cao L, Wang X. Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity. J Korean Math Soc. 2021;58:1449–1460.
- Chen W, Qi S. Direct methods on fractional equations. Discrete Contin Dyn Syst. 2019;39:1269–1310.
- Chen W, Zhu J. Indefinite fractional elliptic problem and Liouville theorems. J Differ Equ. 2016;260:4758–4785.
- Cheng C, Lü Z, Lü Y. A direct method of moving planes for the system of the fractional Laplacian. Pac J Math. 2017;290:301–320.
- Dai W, Liu Z, Lu G. Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space. Potential Anal. 2017;46:569–588.
- Feng Z, Su Y. Lions-type theorem of the fractional Laplacian and applications. Dyn Partial Differ Equ. 2021;18:211–230.
- Li C, Liu C, Wu Z, et al. Non-negative solutions to fractional Laplace equations with isolated singularity. Adv Math. 2020;373:107329. 38 pp.
- Wang P, Dai Z, Cao L. Radial symmetry and monotonicity for fractional H e´non equation in Rn. Complex Var Elliptic Equ. 2015;60:1685–1695.
- Wang P, Niu P. Liouville's theorem for a fractional elliptic system. Discrete Contin Dyn Syst. 2019;39:1545–1558.
- Wang P. Uniqueness and monotonicity of solutions for fractional equations with a gradient term. Electron J Qual Theory Differ Equ. 2021;58:1–19.
- Caffarelli L, Silvestre L. An extension problem related to the fraction Laplacian. Commun Partial Differ Equ. 2007;32:1245–1260.
- Cao L, Chen W. Liouville type theorems for poly-harmonic Navier problems. Discrete Contin Dyn Syst. 2013;33:3937–3955.
- Li D, Zhuo R. An integral equation on half space. Proc Amer Math Soc. 2010;138:2779–2791.
- Lu G, Zhu J. An overdetermined problem in riese-potential and fractional Laplacian. Nonlinear Anal. 2012;75:3036–3048.
- Chen W, Li C, Ou B. Classification of solutions for an integral equation. Commun Pure Appl Math. 2006;59:330–343.
- Chen W, Li C, Ou B. Qualitative properties of solutions for an integral equation. Discrete Contin Dyn Syst. 2005;12:347–354.
- Chen W, Fang Y, Yang R. Loiuville theorems involving the fractional Laplacian on a half space. Adv Math. 2015;274:167–198.
- Chen W, Li C. Maximum principle for the fractional p-Laplacian and symmetry of solutions. Adv Math. 2018;335:735–758.
- Chen W, Li C, Li G. Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions. Calc Var Partial Differ Equ. 2017;56:29.
- Dai W, Liu Z, Wang P. Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation. Commun Contemp Math. 2022;24:2150005.
- Wu L, Niu P. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete Contin Dyn Syst. 2019;39:1573–1583.
- Chen W, Li C, Li Y. A drirect method of moving planes for the fractional Laplacian. Adv Math. 2017;308:404–437.
- Cheng T, Huang G, Li C. The maximum principles for fractional Laplacian equations and their applications. Commun Contemp Math. 2017;19:1750018. 12pp.
- Li C, Chen W. A Hopf type lemma for fractional equations. Proc Amer Math Soc. 2019;147:1565–1575.
- Dai W, Qin G, Wu D. Direct methods for pseudo-relativistic Schrödinger operators. J Geom Anal. 2021;31:5555–5618.
- Guo Y, Peng S. Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations. Z Angew Math Phys. 2021;72:1–20.
- Cao Y, Wu J, Wang L. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete Contin Dyn Syst. 2019;39:1237–1256.
- Han X, Lu G, Zhu J. Characterization of balls in terms of bessel-potential integral equation. J Differ Equ. 2012;252:1589–1602.
- Lei Y, Li C, Ma C. Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations. Cal Var Partial Differ Equ. 2012;45:43–61.
- Wang L, Wang L, Zhou C. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Commun Pure Appl Anal. 2021;20:1241–1261.
- Wu L, Chen W. The sliding methods for the fractional p-Laplacian. Adv Math. 2020;361:106933.
- Li Y, Nirenberg L. On the Hopf lemma. arXiv:0709.3531, 2007.
- Protter MH, Weinberger HF. Maximum principles in differential equations. New York: Springer-Verlag;1984.
- Wang P, Chen W. Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional p-Laplacians; 2022;21:3055–3069.
- Chen W, Li C, Qi S. A Hopf lemma and regularity for fractional p-Laplacians. Discrete Contin Dyn Syst. 2020;40:3235–3252.
- Chen W, Wang P, Niu Y, et al. Asymptotic method of moving planes for fractional parabolic equations. Adv Math. 2021;377:107463. 47 pp.
- Chen W, Wu L. Liouville theorems for fractional parabolic equations. Adv Nonlinear Stud. 2021;21:939–958.
- Gidas B. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv Math. 1981;0:369–402.
- Wang P, Niu P. Symmetric properties of solutions for fully nonlinear nonlocal system. Nonlinear Anal. 2019;187:134–146.