关于平均曲率方程上下解方法的注记.docVIP

关于平均曲率方程上下解方法的注记 潘洪京1, 邢瑞香2 1 华南师范大学数学科学学院，广州 510631 2 中山大学数学与计算科学学院，广州 510275 摘要：本文改进了 Noussair, Swanson and Yang 对于平均曲率方程建立的闸方法。我们去掉 了原始结果中的额外的条件。作为改进结果的一个直接的应用，我们可以提高已有的关于平均 曲率算子的 Liouville-Bratu-Gelfand 型问题的存在性结果的参数范围。 关键词：基础数学，平均曲率方程，上下解方法，闸方法，径向对称解，Gelfand 问题，指数 非线性 中图分类号： O175.2 Remarks on sub-supersolution methods for prescribed mean curvature equations Hongjing Pan1, Ruixiang Xing2 1 2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631 School of Mathematics Computational Science, Sun Yat-sen University, Guangzhou 510275 Abstract: We improve a barrier method for prescribed mean curvature problems built by Noussair, Swanson and Yang. We remove an additional condition from the original results. As a direct application of the improved results, we may improve a well known existence result on the Liouville-Bratu-Gelfand type problem for the mean curvature operator. Key words: Pure mathematics, prescribed mean curvature equation, sub-supersolution methods, barrier methods, radial solutions, Gelfand problem, exponential nonlinearity 0 Introduction In [1], Noussair, Swanson and Yang established a barrier method for prescribed mean curvature problems. The problem under consideration is ?  u ) = λf(x, u), 1 + | u|2 u 0 in ?, u|?? = 0,  x ∈ ?,  (0.1) 基金项目： Supported by NSFC 11001277，RFDP (200805581023) 作者简介： Hongjing Pan(1979-), male, lecturer, major research direction：nonlinear analysis. Correspondence author： Ruixiang Xing(1979-), female, associate professor, major research direction：nonlinear analysis. -1- where λ is a positive constant and ? is a bounded domain in RN, N  2 and ?? C 2,α, satisfying an interior sphere condition. Assumption (F). f ∈ C1(? × [0, +∞)), f(x, t) 0 for t 0, and f(x, t) is nondecreasing in t ∈ [0, t0] for every x ∈ ? and some t0 0. Assumption (A). There exist numbers λ? 0 and λ? λ? such that (0.1) has a superso- lution v and a subsolution w v in ? for all λ ∈ (