Noncompactness for the constant $Q_{2N}$-curvature problem

For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb{S}^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ whe