Axially Symmetric Rigidity for a $Q$-Curvature-Type Equation on $ \mathbb{S}^N $

We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ α\ge\frac{1}{2}$ and $α\not =1$