Symmetry Properties for Positive Solutions to Some Elliptic Equations in Sector Domains with Large Amplitude

In this paper we study symmetry properties for positive solutions of semilinear elliptic equation u+f u = 0 with mixed boundary condition in a spherical sector α R , where α, the amplitude of the sector, is between π and 2π. Under certain conditions on f u , we prove that all positive solutions are radially symmetric about the origin. Unlike well-known results of B. Gidas et al. (1979, Comm. Math. Phys. 68, 209–243) on Dirichlet problem, or early results of Berestycki and Pacella on the same problem for α R with an acute angle, extra conditions on f u are needed, as pointed out early by H. Berestycki and F. Pacella (1989, J. Funct. Anal. 87, 177–211).