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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2305.07249 |
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| _version_ | 1866908884680048640 |
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| author | Lê, Quynh N. T. Ngô, Quôc Anh Nguyen, Tien-Tai |
| author_facet | Lê, Quynh N. T. Ngô, Quôc Anh Nguyen, Tien-Tai |
| contents | 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 \[ {\mathbf P}_n^{2s}(v) = Q_n^{2s}(\varepsilon v+v^α) \] on $\mathbf S^n$ with $0<α\leq (n+2s)/(n-2s)$, 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 $0<α\leq (n+2s)/(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon=0$ but with $0<α< (n+2s)/(n-2s)$.As a by-product, with $0<α\leq (n+2s)/(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities \[ \int_{\mathbf S^n} v {\mathbf P}_n^{2s} (v) dμ_{g_{\mathbf S^n}} \geq \frac{Γ(n/2 + s)}{Γ(n/2 - s )} | \mathbf S^n|^\frac{α-1}{α+1} \Big( \int_{\mathbf S^n} v^{α+1} dμ_{g_{\mathbf S^n}} \Big)^\frac{2}{α+1}. \] for the GJMS operator ${\mathbf P}_n^{2s}$ on $\mathbf S^n$ and for all non-negative functions $v\in H^s(\mathbf S^n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_07249 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Liouville type result for fractional GJMS equations on higher dimensional spheres Lê, Quynh N. T. Ngô, Quôc Anh Nguyen, Tien-Tai Analysis of PDEs 35A15, 35A23, 35C15, 45H05, 58J70 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 \[ {\mathbf P}_n^{2s}(v) = Q_n^{2s}(\varepsilon v+v^α) \] on $\mathbf S^n$ with $0<α\leq (n+2s)/(n-2s)$, 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 $0<α\leq (n+2s)/(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon=0$ but with $0<α< (n+2s)/(n-2s)$.As a by-product, with $0<α\leq (n+2s)/(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities \[ \int_{\mathbf S^n} v {\mathbf P}_n^{2s} (v) dμ_{g_{\mathbf S^n}} \geq \frac{Γ(n/2 + s)}{Γ(n/2 - s )} | \mathbf S^n|^\frac{α-1}{α+1} \Big( \int_{\mathbf S^n} v^{α+1} dμ_{g_{\mathbf S^n}} \Big)^\frac{2}{α+1}. \] for the GJMS operator ${\mathbf P}_n^{2s}$ on $\mathbf S^n$ and for all non-negative functions $v\in H^s(\mathbf S^n)$. |
| title | A Liouville type result for fractional GJMS equations on higher dimensional spheres |
| topic | Analysis of PDEs 35A15, 35A23, 35C15, 45H05, 58J70 |
| url | https://arxiv.org/abs/2305.07249 |