A direct method of moving planes for the fractional Laplacian

Brandle, 2013, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143, 39, 10.1017/S0308210511000175 Caffarelli, 2007, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 1245, 10.1080/03605300600987306 Caffarelli, 2009, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62, 597, 10.1002/cpa.20274 Chen, 2015, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167, 10.1016/j.aim.2014.12.013 Chen, 1991, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 615, 10.1215/S0012-7094-91-06325-8 Chen, 2010, Methods on Nonlinear Elliptic Equations, vol. 4 Chen, 2016, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations Chen, 2005, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12, 347, 10.3934/dcds.2005.12.347 Chen, 2006, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59, 330, 10.1002/cpa.20116 Chen, 2016, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260, 4758, 10.1016/j.jde.2015.11.029 Fang, 2012, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. Math., 229, 2835, 10.1016/j.aim.2012.01.018 Frank, 2013, Uniqueness and nondegeneracy of ground states for (−△)sQ+Q−Qα+1=0 in R, Acta Math., 210, 261, 10.1007/s11511-013-0095-9 Frank Frank, 2016, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69, 1671, 10.1002/cpa.21591 Frank, 2010, Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality, Calc. Var. Partial Differential Equations, 39, 85, 10.1007/s00526-009-0302-x Gidas, 1979, Symmetry and the related properties via the maximum principle, Comm. Math. Phys., 68, 209, 10.1007/BF01221125 Han, 2012, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252, 1589, 10.1016/j.jde.2011.07.037 Hang, 2007, On the integral systems related to Hardy–Littlewood–Sobolev inequality, Math. Res. Lett., 14, 373, 10.4310/MRL.2007.v14.n3.a2 Hang, 2009, An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 1, 10.1016/j.anihpc.2007.03.006 Jarohs, 2016, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195, 273, 10.1007/s10231-014-0462-y Jin, 2014, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16, 1111, 10.4171/JEMS/456 Jin, 2014, A fractional Yemabe flow and some applications, J. Reine Angew. Math., 696, 187, 10.1515/crelle-2012-0110 Lei, 2013, Asymptotic properties of positive solutions of the Hardy–Sobolev type equations, J. Differential Equations, 254, 1774, 10.1016/j.jde.2012.11.008 Lei, 2012, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45, 43, 10.1007/s00526-011-0450-7 Lu, 2011, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253, 455, 10.2140/pjm.2011.253.455 Lu, 2011, Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality, Calc. Var. Partial Differential Equations, 42, 563, 10.1007/s00526-011-0398-7 Lu, 2012, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75, 3036, 10.1016/j.na.2011.11.036 Ma, 2006, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5, 855, 10.3934/cpaa.2006.5.855 Ma, 2010, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455, 10.1007/s00205-008-0208-3 Silvestre, 2007, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60, 67, 10.1002/cpa.20153 Zhuo, 2016, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36, 1125