Abstract :
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).
