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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2307.05401 |
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| _version_ | 1866929368525176832 |
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| author | Hyder, Ali Ngô, Quôc Anh |
| author_facet | Hyder, Ali Ngô, Quôc Anh |
| contents | This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation \[
{\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-α} ) \] on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m $, $0<α\leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here $ {\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<α\leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ and $0<α< (2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $α=7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities \[ \Big( \int_{\mathbb S^n} v^{1-α} dμ_{\mathbb S^n} \Big)^{\frac {2}{α-1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) dμ_{\mathbb S^n} \geq \frac{Γ(n/2 + m)}{Γ(n/2 - m )} | \mathbb S^n|^\frac{α+ 1}{α- 1} \] for the GJMS operator $ {\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $α\in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $α=1$, is also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_05401 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Hang-Yang conjecture for GJMS equations on $\mathbb S^n$ Hyder, Ali Ngô, Quôc Anh Analysis of PDEs Differential Geometry 53C18, 58J05, 35A23, 26D15 This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation \[ {\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-α} ) \] on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m $, $0<α\leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here $ {\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<α\leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ and $0<α< (2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $α=7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities \[ \Big( \int_{\mathbb S^n} v^{1-α} dμ_{\mathbb S^n} \Big)^{\frac {2}{α-1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) dμ_{\mathbb S^n} \geq \frac{Γ(n/2 + m)}{Γ(n/2 - m )} | \mathbb S^n|^\frac{α+ 1}{α- 1} \] for the GJMS operator $ {\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $α\in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $α=1$, is also presented. |
| title | On the Hang-Yang conjecture for GJMS equations on $\mathbb S^n$ |
| topic | Analysis of PDEs Differential Geometry 53C18, 58J05, 35A23, 26D15 |
| url | https://arxiv.org/abs/2307.05401 |