For Hardy-Sobolev inequalities in cones, a sufficiently small first Neumann eigenvalue of the spherical cross-section forces the extremal function to be non-radial.
and Pacella, F., Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone
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Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones
For Hardy-Sobolev inequalities in cones, a sufficiently small first Neumann eigenvalue of the spherical cross-section forces the extremal function to be non-radial.