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

This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation $${\mathbf{P}}_n^{2m}(v)=\frac{n-2m}{2}{Q}_n^{2m}(\epsilon v+v^{-\alpha })$$ on ${\mathbb{S}}^n$ with $m\ge 2$, $3\le n<2m$, $0<\alpha \le (2m+n)\mathrm{/}(2m-n)$, and $\epsilon >0$. Here ${\mathbf{P}}_n^{2m}$ is the GJMS operator of order $2m$ on ${\mathbb{S}}^n$ and $Q_n^{2m}=(2\mathrm{/}(n-2m)){\mathbf{P}}_n^{2m}(1)$ is constant. We show that if $\epsilon >0$ is small and $0<\alpha \le (2m+n)\mathrm{/}(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\epsilon =0$ and $0<\alpha <(2m+n)\mathrm{/}(2m-n)$. 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 $$({\int }_{{\mathbb{S}}^n}{v}^{1-\alpha }d{\mu }_{{\mathbb{S}}^n}{)}^{\frac{2}{\alpha -1}}{\int }_{{\mathbb{S}}^n}v{\mathbf{P}}_n^{2m}(v)d{\mu }_{{\mathbb{S}}^n}\ge \frac{\mathrm{\Gamma }(n\mathrm{/}2+m)}{\mathrm{\Gamma }(n\mathrm{/}2-m)}\mathrm{\mid }{\mathbb{S}}^n{\mathrm{\mid }}^{\frac{\alpha +1}{\alpha -1}}$$ 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.