Cheng–Yau Logarithmic Gradient Estimates for a Nonlinear Elliptic Equation on Smooth Metric Measure Spaces
In this paper, we consider the nonlinear elliptic equation$$\Delta_fv^\tau+\lambda v=0$$on a complete smooth metric measure space with the $m$-Bakry-Émery Ricci curvature bounded from below, where $\tau>0$ and $\lambda$ are constants. We obtain some new local and global universal $\log$-g...
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| Datum: | 2026 |
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| Ключові слова: | keywords |
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| Sprache: | Englisch |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2026
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| Online Zugang: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1131 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Zusammenfassung: | In this paper, we consider the nonlinear elliptic equation$$\Delta_fv^\tau+\lambda v=0$$on a complete smooth metric measure space with the $m$-Bakry-Émery Ricci curvature bounded from below, where $\tau>0$ and $\lambda$ are constants. We obtain some new local and global universal $\log$-gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem, a Harnack inequality and the global gradient estimates for such solutions. Our results generalize and improve the estimates established by Wang (J. Differential Equations 260 (2016), 567--585) and Zhao (Arch. Math. (Basel) 114 (2020), 457-469).
Mathematical Subject Classification 2020: 58J05, 35B45 |
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