Symmetry and asymmetry: The method of moving spheres

Abstract

This paper consists of two parts. The first part concerns a question raised by Véron on the symmetry property of positive solutions of the semilinear elliptic equation. The second part concerns some nonlinear elliptic equations on the unit sphere Sn. By the method of moving spheres and the global bifurcation theory, we obtain various symmetry, asymmetry, and non-existence results.