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

Abstract

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

arXiv comment29 pages

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