Degenerate Solutions of Yamabe-Type Equations on Products of Spheres

This is the authors' abstract. We don't add key points for this paper.

We study the Yamabe-type equation -ΔG_δu+λ u=λ uq-1 on( Sⁿ× Sⁿ,\,G_δ=g₀ +δ g₀), restricting to solutions invariant under the diagonal action of $O(n+1)$. This action has cohomogeneity one, and there exists a smooth isoparametric function f: Sⁿ× Sⁿ→[-1,1] such that every invariant solution can be written as u=φ∘ f for a unique profile φ. We show that the equation reduces to a singular ODE for φ on $[-1,1]$. As a consequence, bifurcation methods provide families of nonconstant solutions that depend nontrivially on both factors. In particular, for every even k we obtain a positive invariant solution at which the linearized operator has nontrivial kernel, hence a degenerate solution.