具有г—变化非线性项的无穷拉普拉斯方程的边界爆破分析.docx

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大连理工大学硕士学位论文
摘 要
本文研究∞一Laplace方程△oo性=6(z)‘厂(u)于Q C Rn的边界爆破解,其中非线性项0≤f∈w1≯【o,o。)且在无穷远处r变化,权函数b∈G(Q)在Q内为正,在边界上可能为零.
:b(在边界附近)衰减到零的速率越快或,(当U趋于无穷大时的)增长速率越慢,解(在边界处),我们给出几个具有不同衰减速率b和增长速率,的实例具体描述这些结论,从而得到关于权函数b(在边界附近)的衰减速率与非线性项。,(在无穷远处)的增长速率如何影响解的边界爆破速率的清晰图画.
,,首先回顾我们另外一篇文章里得到的非线性项是正则变化的无穷拉普拉斯方程的边界爆破结论,然后给出本文非线性项是r变化的无穷拉普拉斯方程的主要结果(即定理1).,在第5章,为得到描述结论的清晰映像,我们列举了一系列满足定理1三种情形要求的函数作为例子.
关键词:无穷拉普拉斯算子;边界爆破;渐近估计;F变化;比较原理
大连理工大学硕士学位论文
Boundary Blow-up in Infinity Laplace Equations with I、一Varying Nonlinearity
Abstract
In this paper we study the boundary blow-up solutions to the infinity Laplace equa- tion A∞札=b(x)f(u)in Q C R札,with the nonlinearity 0冬I∈W?[o,∞)F-varying at
∞,and the weighted function b∈c(a)positive in Q and vanishing on the boundary.
We quantitatively determine the boundary asymptotic behavior of solutions,relying
on the decay rate of b near the boundary and the growth rate of/at is shown that the faster b decays to zero,or the more slowly I grows at(20,the faster the solutions go to infinity near the addition,we characterize these results via examples possessing various decay rates for b and growth rates for I,and thus obtain a clear picture to illustrate the contribution of the decay of the weighted function b(near the boundary)
and the growth of the nonlinearity,(at infinity)to the boundary blow-up rate of the solutions.
Chapter l introduces the background and current progress of the field,and briefly describes the contents of the Chapter 2,we give some definitions and auxil- iary results aJs preliminaries that will be