On the Hang-Yang conjecture for GJMS equations on ${\mathbb{S}}^n$

Abstract

This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation \[ {\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-α} ) \] on ${\mathbb{S}}^n$ with $m\ge 2$, $3\le n<2m$, $0<\alpha \le (2m+n)/(2m-n)$, and $\epsilon >0$. Here ${\mathbf{P}}_n^{2m}$ is the GJMS operator of order $2m$ on ${\mathbb{S}}^n$ and $Q_n^{2m}=(2/(n-2m)){\mathbf{P}}_n^{2m}(1)$ is constant. We show that if $\epsilon >0$ is small and $0<\alpha \le (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\epsilon =0$ and $0<\alpha <(2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $\alpha =7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities \[ \Big( \int_{\mathbb S^n} v^{1-α} dμ_{\mathbb S^n} \Big)^{\frac {2}{α-1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) dμ_{\mathbb S^n} \geq \frac{Γ(n/2 + m)}{Γ(n/2 - m )} | \mathbb S^n|^\frac{α+ 1}{α- 1} \] for the GJMS operator ${\mathbf{P}}_n^{2m}$ on ${\mathbb{S}}^n$ under the conditions $n\ge 3$, $n=2m-1$, and $\alpha \in (0,1)\cup (1,2n+1]$. A log-Sobolev type inequality, as the limiting case $\alpha =1$, is also presented.

31 pages, 2 figures