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

AbstractIn 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