Symmetry properties for positive solutions of elliptic equations with mixed boundary conditions

Abstract

In this paper we establish symmetry results for positive solutions of semilinear elliptic equations of the type Δu + f( u) = 0 with mixed boundary conditions in bounded domains. In particular we prove that any positive solution u of such an equation in a spherical sector ∑( α, R) is spherically symmetric if α, the amplitude of the sector, is such that 0 < α ⩽ π. By constructing counterexamples we show that this result is optimal in the sense that it does not hold for sectors bE( α, R) with amplitude π < α < 2π. More general symmetry properties are established for positive solutions in domains with axial symmetry. These results extend the well-known theorems of B. Gidas, W. M. Ni, and L. Nirenberg [ Comm. Math. Phys. 68 (1979) , 209–243] to sector-like domains and mixed boundary conditions.