Degenerate Solutions of Yamabe-Type Equations on Products of Spheres

We study Yamabe-type equations on the product of two spheres $(S^n\times S^n,G_{\delta })$, where $G_{\delta }$ is a family of Riemannian metrics parametrized by $\delta >0$. Using bifurcation theory and isoparametric functions, we establish the existence of degenerate solutions that are invariant under the diagonal action of $O(n+1)$ and depend non-trivially on both factors. Our analysis relies on the properties of Gegenbauer polynomials and a careful application of local bifurcation techniques for simple eigenvalues. These results extend previous studies by demonstrating the emergence of solutions that do not solely depend on a single factor, thereby providing new insights into the structure of solutions for Yamabe-type problems on product manifolds.