In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M,g)$$\mathrm{\Delta }u+au{(\log u)}^p+bu=0,$ where $a,b\in \mathbb{R}$ and p is a rational number with $p=\frac{k_1}{2k_2+1}\ge 2$ where $k_1$ and $k_2$ are positive integer numbers. we obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on $(M,g)$. Our results can be viewed as a natural extension of Yau's estimates on positive harmonic function.

中文翻译：

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油型梯度估计Δ Ü  +  AU（log⁡ û）^p  +  BU  = 0上黎曼流形

在本文中，我们考虑在完整的黎曼流形上定义的以下方程的正解的梯度估计 $（\mathrm{中号}，G）$$\mathrm{\Delta }ü+\mathrm{一种}ü{（\mathrm{日志}ü）}^p+bü=0，$ 哪里 $\mathrm{一种}，b\in \mathbb{[R}$和p是一个有理数与$p=\frac{{ķ}_{\mathrm{1个}}}{2{ķ}_2+\mathrm{1个}}\ge 2$ 哪里 ${ķ}_{\mathrm{1个}}$ 和 ${ķ}_2$是正整数。我们获得方程正解的梯度边界，该梯度解不取决于解的边界和距离函数的拉普拉斯算子$（\mathrm{中号}，G）$。我们的结果可以看作是Yau对正谐波函数的估计的自然扩展。