Remark on some conformally invariant integral equations: the method of moving spheres

where a > 0 is some constant and x̄ ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [6]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([22]-[25]); see also related works of Viaclovsky ([35]-[36]) and Chang, Gursky and Yang ([11]-[12]). The method used in [19], as well as in much of the above cited work, is the method of moving planes. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations, see Alexandrov [1], Serrin [33], Gidas, Ni and Nirenberg [19]-[20], Berestycki and Nirenberg [2], and others. In [28], Li and Zhu gave a proof of the above mentioned theorem of Caffarelli, Gidas and Spruck using the method of moving spheres (i.e. the method of moving