Gradient estimates for $p$-Laplacian equation with cubic polynomial nonlinearity on Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914361437585408 |
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| author | Qiu, Zhen Wang, Youde Yang, Jun |
| author_facet | Qiu, Zhen Wang, Youde Yang, Jun |
| contents | This paper studies a class of $p$-Laplace equations with cubic polynomial nonlinearity \[ Δ_p v + (v-a_1)(v-a_2)(v-a_3) = 0 \] on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1 < a_2 < a_3$ are real constants and $Δ_p v = \operatorname{div}(|\nabla v|^{p-2}\nabla v)$ denotes the $p$-Laplace operator. Depending on whether the solution lies in the intervals $(a_1,a_2), (a_2,a_3)$ or $(a_1,a_3)$, we employ, respectively, a logarithmic transformation or a hyperbolic tangent transformation to convert the original equation to another one for further analysis. Through a detailed analysis of the lower-bound estimate for the linearized operator of the new equation, and by combining Saloff-Coste's Sobolev inequality with a Moser iteration, we establish Cheng-Yau type gradient estimates under an additional assumption on $p$. As applications, the Liouville theorem and a Harnack inequality are further proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00933 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for $p$-Laplacian equation with cubic polynomial nonlinearity on Riemannian manifolds Qiu, Zhen Wang, Youde Yang, Jun Analysis of PDEs 35J92, 58J05, 35B45 This paper studies a class of $p$-Laplace equations with cubic polynomial nonlinearity \[ Δ_p v + (v-a_1)(v-a_2)(v-a_3) = 0 \] on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1 < a_2 < a_3$ are real constants and $Δ_p v = \operatorname{div}(|\nabla v|^{p-2}\nabla v)$ denotes the $p$-Laplace operator. Depending on whether the solution lies in the intervals $(a_1,a_2), (a_2,a_3)$ or $(a_1,a_3)$, we employ, respectively, a logarithmic transformation or a hyperbolic tangent transformation to convert the original equation to another one for further analysis. Through a detailed analysis of the lower-bound estimate for the linearized operator of the new equation, and by combining Saloff-Coste's Sobolev inequality with a Moser iteration, we establish Cheng-Yau type gradient estimates under an additional assumption on $p$. As applications, the Liouville theorem and a Harnack inequality are further proved. |
| title | Gradient estimates for $p$-Laplacian equation with cubic polynomial nonlinearity on Riemannian manifolds |
| topic | Analysis of PDEs 35J92, 58J05, 35B45 |
| url | https://arxiv.org/abs/2603.00933 |