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Research on the nonlinear elliptic eigenvalue problems by variational methods

非线性椭圆特征值问题的变分法研究

基本信息

批准号：
12640211
负责人：
SHIBATA Tetsutaro
金额：
$ 1.47万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640211/
关键词：
Nonlinear Eigenvalue Asymptotic Analysis Singular Perturbation Variational Methods 楕円型方程式

项目摘要

1. (1) We studied the nonlinear two-parameter problem -u"(x) + λu(x)^q = μu(x)^p,u(x) > 0,x ∈(0,1),u(0) = u(1) = 0. Here 1 < q < p are constants and λ,μ > 0 are parameters. We established precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p = (3q - 1)/2 and this kind of criticality is new.(2) We considered the nonlinear two-parameter problem u"(x) + μu(x)^p = λu(x)^q, u(x) > 0, x ∈ I = (0,1), u(0) = u(1) = 0, where 1 < q < p < 2q + 3 and λ,μ > 0 are parameters. We established the three-term spectral asymptotics for the eigencurve λ = λ(μ) as μ→∞ by using a variational method on the general level set due to Zeidler. The first and second terms of'λ(μ) do not depend on the relationship between p and q. However, the third term depends on the relationship between p and q, and the critical case is p = (3q-1)/2.2. We considered the nonlinear eigenvalue problem -Δu = λf(u), u > 0 in Ω,u = 0 on ?∂Ω, where Ω=B_R={x ∈ R^N : |x| <R} or A_<a,R> = {x ∈ R^N : a< |x| <R} (N【greater than or equal】2) and λ>0 is a parameter. It is known that under some conditions on f and g, the corresponding solution u_λ develops boundary layers when λ>> 1. We established the asymptotic formulas for the width of the boundary layers with exact second term and the estimate of the third term.3. We considered several elliptic partial differential equations and parabolic systems related to nonlinear eigenvalue problems and obtained some existence results and qualitative properties of the solutions.

1.(1)研究了两参数非线性问题-u“(X)+λu(X)^q=μu(X)^p，u(X)&gt；0，x∈(0，1)，u(0)=u(1)=0。这里，1&lt；q&lt；p是常量，λ，μ&gt；0是参数。我们建立了变分特征曲线μ(λ)为λ→∞的精确的渐近公式。(2)我们考虑了两参数非线性问题u“(X)+μu(X)^p=λu(X)^q，u(X)&gt；0，x∈i=(0，1)，u(0)=u(1)=0，其中1&lt；q&lt；p&lt；2q+3和λ，μ&gt；0是参数。我们利用Zeidler在一般水平集上的变分方法，建立了特征曲线λ=λ(μ的三项谱渐近性态为μ→∞。λ(μ)的第一项和第二项不依赖于p和q之间的关系，而第三项取决于p和q之间的关系，临界情况是p=(3q-1)/2.2.我们考虑了非线性特征值问题-Δu=λf(U)，u&gt；0在Ω中，u=0 on？∂Ω，其中Ω=B_R={x∈R^N：|x|&lt；R}或A_&lt；a，R&gt；={x∈R^N：A&lt；|x|&lt；R}(N[大于或等于]2)，λ&gt；0是参数。已知在f和g的某些条件下，相应的解u_λ在λ&gt；&gt；1中发展出边界层。我们建立了边界层宽度的渐近公式和第三项的估计。我们考虑了几个与非线性特征值问题有关的椭圆型偏微分方程组和抛物型方程组，得到了了解的一些存在性结果和定性性质。