Study on Positive Solutions of Nonlinear Elliptic Boundary Value Problems Arising in Population Dynamics

种群动力学中非线性椭圆边值问题的正解研究

• 批准号: 16540165
• 负责人: UMEZU Kenichiro
• 金额: $ 1.73万
• 依托单位: Maebashi Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540165/
• 关键词: Nonlinear elliptic problem; Nonlinear boundary condition; Multiplicity; Bifurcation equation; Principal eigenvalue; Variational technique; Population dynamics; 線形固有値問題; 主固有値の発散; ロバン型境界条件; 空間1次元; 変分的特徴付け; 線形化固有値問題; ノイマン境界条件; 発散; 必要十分条件; 分岐パ

In this research we studied existence and behavior of positive solutions to a nonlinear elliptic boundary value problem of logistic type in a smooth bounded domain of the Euclidean space, as a parameter included varies.1. We established a theorem on bifurcation and stability of positive solutions under nonlinear boundary conditions. A local bifurcation analysis was carried out by using the Lyapunov-Schmidt type of procedure, the local bifurcation theory due to Crandall and Rabinowitz and the method of super and subsolutions. Moreover we reduced our problem to a constrained minimization one based on the first bifurcation solution, to obtain multi plicity of positive solutions.2. We established a nonexistence theorem for positive solutions under nonlinear boundary conditions when parameter is small enough, where a variational technique and super and subsolutions are used.3. We studied a linear indefinite eigenvalue problem with a Robin type boundary condition. Existence and uniqueness of a positive principal eigenvalue was proved by use of a variational approach. We also obtained an a priori lower bound for the positive principal eigenvalue in terms of the sign-changing weight included in the given eigenvalue problem. We remark that we can get the linear eigenvalue problem under consideration if the nonlinear elliptic boundary value problem of logistic type is linearized at the trivial solution which is identically equal to zero.4. We studied blowing-up property of the positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property were discussed. It should be emphasized that we constructed a counterexample to show that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true.

在这项研究中，我们研究了在欧几里得空间的平滑界面域中对逻辑类型的非线性椭圆边界值问题的积极解决方案的存在和行为，因为包括参数随附了。1。我们在非线性边界条件下建立了关于阳性溶液的分叉和稳定性定理。通过使用lyapunov-schmidt类型的程序，由于crandall和Rabinowitz引起的局部分叉理论以及超级和亚种的方法，进行了局部分叉分析。此外，我们将问题降低到基于第一个分叉溶液的约束最小化，以获得阳性溶液的多层次。2。当参数足够小时，我们在非线性边界条件下建立了一个不存在的定理，其中使用变异技术，超级和亚物物。3。我们研究了一个线性不确定的特征值问题，并使用了罗宾类型边界条件。通过使用各种方法证明了积极主征价的存在和独特性。我们还根据给定特征值问题所包含的签名重量来获得正主特征值的先验下限。我们指出的是，如果在琐事解决方案处将非线性椭圆边界值问题线性化，我们可以考虑正在考虑的线性特征值问题，而非线性椭圆形的边界值问题与零相同。4。我们研究了正主特征值的吹吹，用于无限重量和诺伊曼边界条件的线性椭圆特征值问题。讨论了爆破财产的必要条件。应该强调的是，我们构建了一个反例，以表明在Dirichlet边界条件情况下，已知的已知必要条件不再是真实的。

项目成果

Multiplicity of positive solutions under nonlinear boundry conditions for diffusive logistic equations

扩散Logistic方程非线性边界条件下的正解重数

• 发表时间: 2004
• 期刊: Proceedings of the Edinburgh Mathematical Society 47巻2号
• 作者: K.Watanabe;T.Kobayashi;T.Suzuki;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu
• 通讯作者: Kenichiro Umezu

MULTIPLICITY OF POSITIVE SOLUTIONS UNDER NONLINEAR BOUNDARY CONDITIONS FOR DIFFUSIVE LOGISTIC EQUATIONS

• DOI: 10.1017/s0013091503000294
• 发表时间: 2004-06
• 期刊: Proceedings of the Edinburgh Mathematical Society
• 影响因子: 0.7
• 作者: K. Umezu

Local bifurcation analysis and stability of steady-state solutions for diffusive logistic equations with nonlinear boundary conditions

非线性边界条件扩散Logistic方程的局部分岔分析和稳态解的稳定性

• 发表时间: 2004
• 期刊: Communications in Applied Analysis 8(4)
• 作者: K.Watanabe;T.Kobayashi;T.Suzuki;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu
• 通讯作者: Kenichiro Umezu

One parameter-dependent nonlinear elliptic boundary value problems arising in population dynamics

种群动态中出现的一种参数相关的非线性椭圆边值问题

• 发表时间: 2005
• 期刊: in the book "Advances in analysis", World Sci.Publ
• 作者: K.Watanabe;T.Kobayashi;T.Suzuki;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu;Kenichiro Umezu
• 通讯作者: Kenichiro Umezu

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

• DOI: 10.1080/00036810500337860
• 发表时间: 2006-11
• 期刊: Applicable Analysis
• 影响因子: 1.1
• 作者: K. Umezu