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 \[ {\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-\alpha} ) \] on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m $, $0<\alpha \leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here $ {\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<\alpha \leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ 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 \[ \Big( \int_{\mathbb S^n} v^{1-\alpha} d\mu_{\mathbb S^n} \Big)^{\frac {2}{\alpha -1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) d\mu_{\mathbb S^n} \geq \frac{\Gamma (n/2 + m)}{\Gamma (n/2 - m )} | \mathbb S^n|^\frac{\alpha + 1}{\alpha - 1} \] for the GJMS operator $ {\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $\alpha \in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $\alpha=1$, is also presented.