DOI: 10.36890/iejg.1862184 ISSN: 1307-5624

Degenerate Solutions of Yamabe-Type Equations on Products of Spheres

Héctor Barrantes González
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We study the Yamabe-type equation $-\Delta_{G_\delta}u+\lambda u=\lambda u^{q-1}$ on$(\mathbb S^n\times\mathbb S^n,\,G_\delta=g_0 +\delta g_0)$, restricting to solutions invariant under the diagonal action of $O(n+1)$. This action has cohomogeneity one, and there exists a smooth isoparametric function $f:\mathbb S^n\times\mathbb S^n\to[-1,1]$ such that every invariant solution can be written as $u=\phi\circ f$ for a unique profile $\phi$. We show that the equation reduces to a singular ODE for $\phi$ 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.

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