半线性椭圆方程边界爆破解的定性性质 Alternative Title Qualitative properties of boundary blow-up solutions to semilinear elliptic equations 黄水波 Thesis Advisor 李万同 2013-05-26 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 边界爆破解 唯一性 渐近行为 Karamata 正规变化理论 上, 下解方法 Abstract 本文的主要目的是研究非线性椭圆方程解的唯一性和边界行为。 其中区域是有界光滑区域, 权函数非负且在边界附近可能奇异或趋于零, f局部Lipschitz连续且f(u)/u在递增. 类似的方程来源于诸多领域，近几十年吸引了大批学者的研究兴趣. 首先, 在f和b(x)的满足各种假设条件下, 建立了方程边界爆破解的唯一性和渐近行为. 具体来说, 首先, 当f，b(x)具有精确渐近行为时, 建立了方程边界爆破解的唯一性和渐近行为. 该结论推广了Ouyang和Xie的一系列结果。其次, 引入新的函数类并建立了方程边界爆破解的渐近行为.随后, 建立了边界爆破解渐近行为的统一且显式的公式。 最后, 着重处理b(x)在边界附近的衰减速率与f在无穷远处的增长速率之间的竞争.对 Cirstea本人对该问题做了部分解决. 我们也将给出此公开问题的部分解答. 其次, 我们考虑了区域的几何性质对边界爆破解的二阶估计的影响. 众所周知, 边界爆破解的一阶渐近行为与区域的几何性质无关, 但是二阶渐近行为依赖于边界的平均曲率. 据我们所知,已有的结果都是在特殊函数的情形下得到的. 我们建立了边界爆破解的二阶渐近行为. 最后, 我们建立了距离函数对边界爆破解的二阶估计的影响. Other Abstract The purpose of this thesis is to investigate a class of nonlinear elliptic equations.b(x) is non-negative for some, f is locally Lipschitz continuous and f(u)/u is increasing . Firstly, we treat the uniqueness and asymptotic behaviour of boundary blow-up solutions under different conditions on f and b(x). More precisely, firstly, we consider the uniqueness and asymptotic behaviour of boundary blow-up solutions f and b(x) have exact asymptotic behaviour. This results extends a series of works of Ouyang and Xie. Secondly, we introduce a new class functions and obtain the boundary asymptotic behaviour. Next, we establish a unified and explicit asymptotic formula of boundary blow-up solutions. Finally, we deal with the competition between the vanishing rate of b(x) at boundary and the growth rate of f near infinity. Secondly, the effect of the mean curvature of the nearest point on the boundary in the second order approximation of the boundary blow-up solution is discussed. Finally, we establish the second order approximation of boundary blow-up solutions which depends on the distance functions. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225624 Collection 数学与统计学院 Recommended CitationGB/T 7714 黄水波. 半线性椭圆方程边界爆破解的定性性质[D]. 兰州. 兰州大学,2013.
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