Liouville theorem for the inequality $Δ_m u+f(u)\leq 0$ on Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916958996267008 |
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| author | Zhao, Biqiang |
| author_facet | Zhao, Biqiang |
| contents | In this paper, we study the quasilinear inequality $ Δ_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,α>m-1 \quad and \quad f(t)> 0,αf(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some point $x_0$ and large enough $r$, \begin{align*} vol B_r(x_0)\leq C r^p ln^q r, \end{align*} where $p=\frac{mα}{α-(m-1)},q=\frac{m-1}{α-(m-1)}$ and $B_r(x_0) $ is a geodesic ball of radius $r$ centered at $x_0$, then the inequality possesses no positive weak solution. This generalizes the result in \cite{AS,Sun}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville theorem for the inequality $Δ_m u+f(u)\leq 0$ on Riemannian manifolds Zhao, Biqiang Analysis of PDEs Differential Geometry In this paper, we study the quasilinear inequality $ Δ_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,α>m-1 \quad and \quad f(t)> 0,αf(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some point $x_0$ and large enough $r$, \begin{align*} vol B_r(x_0)\leq C r^p ln^q r, \end{align*} where $p=\frac{mα}{α-(m-1)},q=\frac{m-1}{α-(m-1)}$ and $B_r(x_0) $ is a geodesic ball of radius $r$ centered at $x_0$, then the inequality possesses no positive weak solution. This generalizes the result in \cite{AS,Sun}. |
| title | Liouville theorem for the inequality $Δ_m u+f(u)\leq 0$ on Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2509.16659 |