Let n be an integer and s be a real number such that \(n > 2s \ge 2\) . Inspired by the perturbation approach initiated by Hang and Yang (Int. Math. Res. Not. IMRN, 2020), we are interested in non-negative, smooth solution v to the following higher-order fractional equation \(\begin{aligned} \, {\textbf{P}}_n^{2s}(v) = \, Q_n^{2s}(\varepsilon v+v^\alpha ) \end{aligned}\) on \(\mathbb {S}^n\) with \(0<\alpha \le (n+2s)/(n-2s)\) , and \(\varepsilon \ge 0\) . Here \(\, {\textbf{P}}_n^{2s}\) is the fractional GJMS type operator of order 2s on \(\mathbb {S}^n\) and \(\, Q_n^{2s}=\, {\textbf{P}}_n^{2s}(1)\) is constant. We show that if \(\varepsilon >0\) and \(0<\alpha \le (n+2s)/(n-2s)\) , 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 \(0<\alpha \le (n+2s)/(n-2s)\) , we compute the sharp constant of the subcritical/critical Sobolev inequalities \(\begin{aligned} \int _{\mathbb {S}^n} v \, {\textbf{P}}_n^{2s}(v) d\mu _{g_{\mathbb {S}^n}} \ge \frac{\Gamma (n/2 + s)}{\Gamma (n/2 - s )} | \mathbb {S}^n|^\frac{\alpha -1}{\alpha +1} \Big ( \int _{\mathbb {S}^n} v^{\alpha +1} d\mu _{g_{\mathbb {S}^n}} \Big )^\frac{2}{\alpha +1} \end{aligned}\) for the GJMS operator \(\, {\textbf{P}}_n^{2s}\) on \(\mathbb {S}^n\) and for all non-negative functions \(v\in H^s(\mathbb {S}^n)\) .
