In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the p-Laplacian$$\begin{aligned} \Delta _{p}u=-\lambda |u|^{p-2}u \end{aligned}$$with \(p>1\) on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the p-Laplacian.

中文翻译：

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p -Laplacian的第一个特征值的上界

在本文中，我们针对p -Laplacian $$ \ begin {aligned} \ Delta _ {p} u =-\ lambda | u | ^ {p-2} u建立以下方程式正解的梯度估计在给定的完整黎曼流形上，\ end {aligned} $$与\（p> 1 \）。因此，我们得出了p -Laplacian的第一个非平凡特征值的上限估计。