Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909983357009920 |
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| author | Jia, Yike |
| author_facet | Jia, Yike |
| contents | In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$Δ_{V}u^{m}+μ(x)u+p(x)u^α=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-Émery curvature is bounded from below by $-(k-1)K$ with $K \geq 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $μ=0$ the equation becomes $Δu^{m}+p(x)u^α=0$. And $V=f$, $μ=c, p=0 $, the equation becomes $Δ_{f}u^{m}+cu=0 $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_03721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian Jia, Yike Analysis of PDEs Differential Geometry 58J35, 35J05 In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$Δ_{V}u^{m}+μ(x)u+p(x)u^α=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-Émery curvature is bounded from below by $-(k-1)K$ with $K \geq 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $μ=0$ the equation becomes $Δu^{m}+p(x)u^α=0$. And $V=f$, $μ=c, p=0 $, the equation becomes $Δ_{f}u^{m}+cu=0 $. |
| title | Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian |
| topic | Analysis of PDEs Differential Geometry 58J35, 35J05 |
| url | https://arxiv.org/abs/2601.03721 |