Abstract: Let M be an n-dimensional complete noncompact Riemannian manifold with metric g, triangle_p(1<pleqslant 2) the p-Laplace operator, by using the classical method of Li-Yau, a gradient estimate of the positive solution to equation triangle_pu=-lambda |u|^p-2u was proved under suitable curvature condition, in which lambdageqslant 0 is a constant; the upper bound estimate of lambda was a byproduct; one also showed that this estimate is sharp. This result generalizes the gradient estimate of the positive solution to elliptic equation triangle u=-lambda
u.

Key words: gradient estimate, sharp estimate, p-Laplace operator, gradient estimate, sharp estimate