Rigidity of a $Q$-curvature-type problem on ${\mathbb{S}}^N$ in every dimension $N\ge 3$

We prove that for every $N\ge 3$, every solution to the $Q$-curvature-type problem $$\alpha P_{N}u+(N-1)!(1-\frac{e^{Nu}}{{\int }_{{\mathbb{S}}^N}{e}^{Nu}dw})=0on{\mathbb{S}}^N$$ is constant, provided that $\alpha \ge \frac{1}{2}$ and $\alpha ≠1$. The proof consists of a spectral gap estimate and pointwise estimates of the normalized density ${\left({\int }_{{\mathbb{S}}^N}{e}^{Nu}dw\right)}^{-1}{e}^{Nu}$.