A Liouville type result for fractional GJMS equations on higher dimensional spheres

Let $n$ be an integer and $s$ be a real number such that $n>2s\ge 2$. Inspired by the perturbation approach initiated by F. Hang and P. Yang ($\mathit{\text{Int.}}\mathit{\text{Math.}}\mathit{\text{Res.}}\mathit{\text{Not.}}\mathit{\text{IMRN}}$, 2020), we are interested in non-negative, smooth solution $v$ to the following higher-order fractional equation $${\mathbf{P}}_n^{2s}(v)=Q_n^{2s}(\epsilon v+v^{\alpha })$$ on ${\mathbf{S}}^n$ with $0<\alpha \le (n+2s)\mathrm{/}(n-2s)$, and $\epsilon \ge 0$. Here ${\mathbf{P}}_n^{2s}$ is the fractional GJMS type operator of order $2s$ on ${\mathbf{S}}^n$ and $Q_n^{2s}={\mathbf{P}}_n^{2s}(1)$ is constant. We show that if $\epsilon >0$ and $0<\alpha \le (n+2s)\mathrm{/}(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\epsilon =0$ but with $0<\alpha <(n+2s)\mathrm{/}(n-2s)$.As a by-product, with $0<\alpha \le (n+2s)\mathrm{/}(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities $${\int }_{{\mathbf{S}}^n}v{\mathbf{P}}_n^{2s}(v)d{\mu }_{g_{{\mathbf{S}}^n}}\ge \frac{\mathrm{\Gamma }(n\mathrm{/}2+s)}{\mathrm{\Gamma }(n\mathrm{/}2-s)}\mathrm{\mid }{\mathbf{S}}^n{\mathrm{\mid }}^{\frac{\alpha -1}{\alpha +1}}({\int }_{{\mathbf{S}}^n}{v}^{\alpha +1}d{\mu }_{g_{{\mathbf{S}}^n}}{)}^{\frac{2}{\alpha +1}}\mathrm{.}$$ for the GJMS operator ${\mathbf{P}}_n^{2s}$ on ${\mathbf{S}}^n$ and for all non-negative functions $v\in H^s({\mathbf{S}}^n)$.