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

Abstract

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

arXiv comment10 pages