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We prove nonuniqueness results for constant sixth order $Q$-metrics oncomplete locally conformally flat $n$-dimensional Riemannian manifolds with$n\geqslant 7$. More precisely, assuming a positive Green function exists forthe sixth order GJMS operator, our objective is two-fold. First, we use aclassical bifurcation technique to prove that there exists infinitely manyconstant $Q$-curvature metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$. As aby-product, we find the sixth order Yamabe invariant on this product manifoldcan be arbitrarily close to that of the round dimensional sphere, generalizinga result of Schoen about the classical Yamabe invariant. Second, when theunderlying manifold is noncompact, we apply a bifurcation technique onRiemannian covering to construct infinitely many complete metrics with constantsixth order $Q$-curvature conformal to $\mathbb{S}^{n_1} \times\mathbb{R}^{n_2}$ or $\mathbb{S}^{n_1} \times \mathbb{H}^{n_2}$, where$n_1+n_2\geqslant 7$. Consequently, we obtain infinitely many solutions to thesingular constant GJMS equation on round spheres $\mathbb{S}^n\setminus\mathbb{S}^k$ blowing up along a minimal equatorial subsphere with $0 \leqslantk<\frac{n-6}{2}$; this dimension restriction is sharp in the topological sense.

Nonuniqueness results for constant sixth order $Q$-curvature metrics on spheres with higher dimensional singularities