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摘要: M是带度量 g 的 n 维非紧黎曼流形,1pleqslant 2 给定常数,triangle_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triangle_pu=-lambda|u|^p-2u 的正函数的一个梯度估计, 其中 lambdageqslant 0是常数; 同时得到了lambda 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triangle 的椭圆方程 triangle u=-lambda u 梯度估计的结果.Abstract: Let M be an n-dimensional complete noncompact Riemannian manifold with metric g, triangle_p(1pleqslant 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.
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Key words:
- p-Laplace operatorgradient estimatesharp estimate /
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