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

Abstract

We prove that for every even integer $N\ge 4$ 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 $\alpha \ge \frac{1}{2}$ and $\alpha \ne 1$. The proof starts from a Gegenbauer expansion and its weighted ${\ell }^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\lfloor G{\rfloor }^2$ using a specially designed integration-by-parts identity for general even integer $N\ge 4$. For $N\in \{3,5,7\}$, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.