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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.10344 |
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| _version_ | 1866910781586538496 |
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| author | Shen, Bin Zhu, Yuhan |
| author_facet | Shen, Bin Zhu, Yuhan |
| contents | In this manuscript, we employ the Nash-Moser iteration technique to determine a condition under which the positive solution $u$ of the generalized nonlinear Poisson equation $$\operatorname{div} (φ(|\nabla u|^2)\nabla u) + ψ(u^2)u = 0,$$ on a complete Riemannian manifold with Ricci curvature bounded from below can be shown to satisfy a Cheng-Yau-type gradient estimate. We define a class of $φ$-Laplacian operators by $Δ_φ(u):=\operatorname{div} (φ(|\nabla u|^2)\nabla u)$, where $φ$ is a $C^2$ function under some certain growth conditions. This can be regarded as a natural generalization of the $p$-Laplacian, the $(p,q)$-Laplacian and the exponential Laplacian, as well as having a close connection to the prescribed mean curvature problem. We illustrate the feasibility of applying the Nash-Moser iteration for such Poisson equation to get the Cheng-Yau-type gradient estimates in different cases with various $φ$ and $ψ$. Utilizing these estimates, we proves the related Harnack inequalities and a series of Liouville theorems. Our results can cover a wide range of quasilinear Laplace operator (e.g. $p$-Laplacian for $φ(t)=t^{p/2-1}$), and Lichnerowicz-type nonlinear equations (i.e. $ψ(t) = At^{p} + Bt^{q} + Ct\log t + D$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Feasibility of Nash-Moser iteration for Cheng-Yau-type gradient estimates of nonlinear equations on complete Riemannian manifolds Shen, Bin Zhu, Yuhan Differential Geometry In this manuscript, we employ the Nash-Moser iteration technique to determine a condition under which the positive solution $u$ of the generalized nonlinear Poisson equation $$\operatorname{div} (φ(|\nabla u|^2)\nabla u) + ψ(u^2)u = 0,$$ on a complete Riemannian manifold with Ricci curvature bounded from below can be shown to satisfy a Cheng-Yau-type gradient estimate. We define a class of $φ$-Laplacian operators by $Δ_φ(u):=\operatorname{div} (φ(|\nabla u|^2)\nabla u)$, where $φ$ is a $C^2$ function under some certain growth conditions. This can be regarded as a natural generalization of the $p$-Laplacian, the $(p,q)$-Laplacian and the exponential Laplacian, as well as having a close connection to the prescribed mean curvature problem. We illustrate the feasibility of applying the Nash-Moser iteration for such Poisson equation to get the Cheng-Yau-type gradient estimates in different cases with various $φ$ and $ψ$. Utilizing these estimates, we proves the related Harnack inequalities and a series of Liouville theorems. Our results can cover a wide range of quasilinear Laplace operator (e.g. $p$-Laplacian for $φ(t)=t^{p/2-1}$), and Lichnerowicz-type nonlinear equations (i.e. $ψ(t) = At^{p} + Bt^{q} + Ct\log t + D$). |
| title | Feasibility of Nash-Moser iteration for Cheng-Yau-type gradient estimates of nonlinear equations on complete Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2405.10344 |