Liouville theorems and new gradient estimates for positive solutions to $Δ_pv+a(v+b)^q=0$ on a complete manifold
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2025
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| author | Wang, Youde Zhang, Linqin |
| author_facet | Wang, Youde Zhang, Linqin |
| contents | In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation $Δ_pv+a(v+b)^q=0$ defined on a complete Riemannian manifold $\left(M,g\right)$ with Ricci lower bound, where $p>1$ is a constant and $Δ_pv=\mathrm{div}\left(\left|\nabla v\right|^{p-2}\nabla v\right)$ is the usual $p$-Laplace operator. Under certain assumptions on $a$, $p$ and $q$, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on $a$, $b$, $p$ and $q$ we show whether or not the exact Cheng-Yau $\log$-gradient estimates for the positive solutions to $Δ_pv+av^q=0$ on $\left(M,g\right)$ with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau $\log$-gradient estimates are established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville theorems and new gradient estimates for positive solutions to $Δ_pv+a(v+b)^q=0$ on a complete manifold Wang, Youde Zhang, Linqin Analysis of PDEs In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation $Δ_pv+a(v+b)^q=0$ defined on a complete Riemannian manifold $\left(M,g\right)$ with Ricci lower bound, where $p>1$ is a constant and $Δ_pv=\mathrm{div}\left(\left|\nabla v\right|^{p-2}\nabla v\right)$ is the usual $p$-Laplace operator. Under certain assumptions on $a$, $p$ and $q$, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on $a$, $b$, $p$ and $q$ we show whether or not the exact Cheng-Yau $\log$-gradient estimates for the positive solutions to $Δ_pv+av^q=0$ on $\left(M,g\right)$ with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau $\log$-gradient estimates are established. |
| title | Liouville theorems and new gradient estimates for positive solutions to $Δ_pv+a(v+b)^q=0$ on a complete manifold |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.12022 |