Abstract
In this paper we study symmetry properties for positive solutions of semilinear elliptic equation Δu+f(u)=0 with mixed boundary condition in a spherical sector Σ(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 Σ(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).
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