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We consider semilinear elliptic equations with mixed boundary conditions inspherical sectors inside a cone. The aim of the paper is to show that a radialsymmetry result of Gidas-Ni-Nirenberg type for positive solutions does not holdin general nonconvex cones. This symmetry breaking result is achieved bystudying the Morse index of radial positive solutions and analyzing how itdepends on the domain D on the unit sphere which spans the cone. In particularit is proved that the Neumann eigenvalues of the Laplace Beltrami operator on Dplay a role in computing the Morse index. A similar breaking of symmetry resultis obtained for the positive solutions of the critical Neumann problem in thewhole unbounded cone. In this case it is proved that the standard bubbles,which are the only radial solutions, become unstable for a class of nonconvexcones.

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones