[1] Gidas B, Ni W M, Nirenberg L. Symmetry and the related properties via the maximum principle. Comm Math Phys, 1979, 68:209-243
[2] Berestycki H, Nirenberg L. Monotonicity. Symmetry and antisymmetry of solutions of semilinear elliptic equations. J Geometry and Physics, 1988, 5:237-275
[3] Gidas B, Ni W M, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv in Math Suppl Stud, 1981, 7A:369-402
[4] Chen W X, Li C M. Classification of solutions of some nonlinear elliptic equations. Duke Math, 1991, 63:615-622
[5] Li Y, Ni W M. On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in Rn. Ⅱ. Radial symmetry. Arch Rational Mech Anal, 1992, 118:223-243
[6] Guo Y X. Non-existence, monotonicity for positive solutions of semilinear elliptic system in R+n. Commun Contemp Math, 2010, 12:351-372
[7] Bianchi G. Non-existence of positive solutions to semilinear elliptic equations on Rn or R+n through the method of moving planes. Comm Partial Differential Equations, 1997, 22:1671-1690
[8] Damascelli L, Gladiali F. Some nonexistence results for positive solutions of elliptice quations in unbounded domains. Rev Mat Iberoamericana, 2004, 20:67-86
[9] Zhang L Z. Symmetry of solutions to semilinear equations involving the fractional Laplacian. Commun Pure Appl Anal, 2015, 14:2393-2409
[10] Fall M M, Weth T. Monotonicity and nonexistence results for some fractional elliptic problems in the half space. Commun Contemp Math, 2016, 18(1):155012, 25pp
[11] Chen W X, Fang Y Q, Yang R. Liouville theorems involving the fractional Laplacian on a half space. Adv. Math, 2015, 274:167-198
[12] Chen W X, Li C M, Li Y. A direct method of moving planes for the fractional Laplacian. arXiv:1411.1697
[13] Zhuo R, Chen W X, Cui X W, Yuan Z X. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete Contin Dyn Syst, 2016, 36:1125-1141
[14] Chen W X, D'Ambrosio L, Li Y. Some Liouville theorems for the fractional laplacian. Nonlinear Anal, 2015, 121:370-381
[15] Zhuo R, Chen W X, Cui X W, Yuan Z X. A Liouville Theorem for the Fractional Laplacian. arXiv:1401.7402
[16] Li Y. Nonexistence of positive solutions for a semi-linear equation involving the fractional Laplacian in Rn. Acta Mathematica Scientia, 2016, 36:666-682
[17] Bogdan K. The boundary Harnack principle for the fractional Laplacian. Studia Math, 1997, 123:43-80
[18] Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60:67-112 |
.jpg){kind=small}
where 0 < α < 2, R+n:={x=(x1, x2,…, xn)|xn > 0}. When K is strictly decreasing with respect to|x'|, the symmetry of positive solutions is proved, where x'=(x1, x2, …, xn-1) ∈ Rn-1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.