一类椭圆方程的梯度估计

Gradient estimates for some elliptic equation

  • 摘要: 通过计算与推导,得到了 n 维完备黎曼流形上椭圆方程 \Delta _gu+au^q\left(\mathrmln\;u\right)^p+bu=0 正解的梯度估计,其不依赖于解的界和距离函数的拉普拉斯(Laplace)算子;将文献1中对正调和函数的梯度估计推广到更一般的情形;将文献10中对一类椭圆方程正解的梯度估计进行了拓展,得到了更具一般性的结果.

     

    Abstract: In this paper, we obtain the gradient estimates of the positive solutions to the following equation defined on an n-dimensional complete Riemannian manifold \Delta _gu+au^q\left(\mathrml\mathrmn\;u\right)^p+bu=0 . The gradient bound does not depend on the bounds of the solution and the Laplacian of the distance function. Our result is an extension of the estimates on positive harmonic function 1 and the estimates of solutions to some nonlinear elliptic equation 10.

     

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