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 $n > 2s \geq 2$. Inspired by the perturbation approach initiated by F. Hang and P. Yang (\textit{Int. Math. Res. Not. IMRN}, 2020), we are interested in non-negative, smooth solution $v$ to the following higher-order fractional equation $${\mathbf P}_n^{2s}(v) = Q_n^{2s}(\varepsilon v+v^\alpha)$$ on $\mathbf S^n$ with $0<\alpha \leq (n+2s)/(n-2s)$, and $\varepsilon \geq 0$. 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 $\varepsilon >0$ and $0<\alpha \leq (n+2s)/(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon=0$ but with $0<\alpha < (n+2s)/(n-2s)$.As a by-product, with $0<\alpha\leq (n+2s)/(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities $$\int_{\mathbf S^n} v {\mathbf P}_n^{2s} (v) d\mu_{g_{\mathbf S^n}} \geq \frac{\Gamma (n/2 + s)}{\Gamma (n/2 - s )} | \mathbf S^n|^\frac{\alpha-1}{\alpha+1} \Big( \int_{\mathbf S^n} v^{\alpha+1} d\mu_{g_{\mathbf S^n}} \Big)^\frac{2}{\alpha+1}.$$ for the GJMS operator ${\mathbf P}_n^{2s}$ on $\mathbf S^n$ and for all non-negative functions $v\in H^s(\mathbf S^n)$.