中文摘要：

M是带度量g的n维非紧黎曼流形，1〈P≤2给定常数，△p是M上的p-Laplace算子，借助于经典的Li—Yau的方法证明了在一定的曲率条件下，满足方程△pu=-λ｜u｜^p-2u的正函数的一个梯度估计，其中A≥0是常数；同时得到了A的一个上界估计；进一步说明了此估计是最优的．推广了关于Laplace算子△的椭圆方程△u=-λu梯度估计的结果．

英文摘要：

Let M be an n-dimensional complete noncompact Riemannian manifold with metric g, △p（1〈P≤2） the p-Laplace operator, by using the classical method of Li-Yau, a gradient estimate of the positive solution to equation △pu=-λ｜u｜^p-2u was proved under suitable curvature condition, in which λ≥ 0 is a constant; the upper bound estimate of was a byproduct; one also showed that this estimate is sharp. This result generalizes the gradient estimate of the positive solution to elliptic equation △u=-λu.