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引用次数: 0
摘要
我们得到了 n 维完整黎曼流形 $(M. g)$$ 上非线性椭圆方程正解的梯度估计值、g)$ $$ \Delta u +au(\ln{u})^{p}+bu\ln{u}=0, $$ 其中 $a\ne 0$ , b 是两个常数,$p=\frac{k_{1}}{2k_{2}+1}\ge 2$ , 这里 $k_{1}$ 和 $k_{2}$ 是两个正整数。梯度边界与解的边界和距离函数的拉普拉奇无关。作为估计值的应用,我们展示了哈纳克不等式和解的上界。
Gradient estimates for a class of elliptic equations with logarithmic terms
We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold $(M, g)$ $$ \Delta u +au(\ln{u})^{p}+bu\ln{u}=0, $$ where $a\ne 0$ , b are two constants and $p=\frac{k_{1}}{2k_{2}+1}\ge 2$ , here $k_{1}$ and $k_{2}$ are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.
期刊介绍:
The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.