Symmetry and Asymmetry: the Method of Moving Spheres

Through the work of Obata [25], Gidas-Ni-Nirenberg [10] and Caffarelli-GidasSpruck [6], the asymptotic behavior of solutions of (1.1) as well as the classification of global solutions are well understood in the case when c = 0. In [28] Véron raised the following question: For c ∈ R, c 6= 0 and n ≥ 3, let u ∈ C∞(Rn\{0}) satisfy (1.1). Is it true that u must be radially symmetric about the origin? He pointed out that there might be non-radial solutions of certain form as suggested in section 4 of [3]. The following result partially answers this question.