Studies on Eigenvalue Problems of Nonlinear Elliptic Equations

非线性椭圆方程特征值问题的研究

• 批准号: 10640208
• 负责人: USAMI Hiroyuki
• 金额: $ 1.15万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640208/
• 关键词: eigenvalue problem; elliptic equation; positive solution; 準線形常微分方程式; 非線形; 楕円形方程式; 固有値

(1) Eigenvalue Problems of Elliptic Equations : Two-parameter eigenvalue problems for semilinear elliptic equations are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Two-parameter Ambrosetti-Prodi problems are also studied. We investigate the relation between parameters and the number of solutions.(2) Positive Solutions of Elliptic Equations : Semilinear second-order elliptic euations are considered in unbounded domains. We establish multiplicity results for positive solutions and uniqueness theorems for positive solutions.(3) Positive Solutions of Quasilinear Ordinary Differential Equations : Quasilinear ordinary differential equations whose leading term is one-dimensionai pseudo-Laplacian are considered. We obtain asynrptotic representations of positive solutions. As an application of these results, we show existence of several types of positive solutions of exterior Dirichlet problems for quasilinear elliptic equations.(4) Mathematical Models Describing Aggregation Phenomena of Molds : We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We clarify the relation between parameters and the number of self-similar solutions.(5) Nonnegative Nontrivial Solutions of Quasilinear Elliptic Equations and Elliptic Systems : We establish necessary and/or sufficient conditions for quasilinear elliptic equations, as well as quasilinear elliptic systems, to possess nontrivial nonnegative entire solutions. Several Liouville type theorems are also obtained.

(1)椭圆方程的特征值问题：研究半线性椭圆方程的二参数特征值问题。我们建立（变分）特征值和特征函数的渐近性质。还研究了双参数 Ambrosetti-Prodi 问题。我们研究了参数与解数之间的关系。(2)椭圆方程的正解：在无界域中考虑半线性二阶椭圆方程。我们建立了正解的重数结果和正解的唯一性定理。(3)拟线性常微分方程的正解：考虑首项为一维伪拉普拉斯算子的拟线性常微分方程。我们获得正解的渐进表示。作为这些结果的应用，我们证明了拟线性椭圆方程的外狄利克雷问题的几种类型的正解的存在性。（4）描述模具聚集现象的数学模型：我们考虑由 Keller 和 Segel 引入的抛物线系统的自相似解来描述由于趋化性引起的模具聚集现象。我们阐明了参数与自相似解的数量之间的关系。(5)拟线性椭圆方程和椭圆系统的非平凡非平凡解：我们建立了拟线性椭圆方程以及拟线性椭圆系统拥有非平凡非负整解的必要和/或充分条件。还得到了几个刘维尔型定理。

项目成果

S. Adachi: "Four positive solutions for the semilinear elliptic equation : -Δμ + μ = a(x)μィイD1ρィエD1 + f(x) in RィイD1NィエD1."Calculus of Variations and Partial Differential Equations. (to appear).

S. Adachi：“半线性椭圆方程的四个正解：-Δμ + μ = a(x)μIID1ρieD1 + f(x) in RiiiD1NieD1。”变分和偏微分方程的微积分。

S.Adachi: "Four positive solutions for the semilinear elliptic equations:-△u+u=a(x)u^p+f(x) in R^N"Calculus of Variations and Partial Differential. (印刷中).

S.Adachi：“半线性椭圆方程的四个正解：R^N 中的-△u+u=a(x)u^p+f(x)”变分和偏微分（正在出版）。

Y.Naito: "Oscillation criteria for quasilinear elliptic equations"Nonlineat Anal.. (印刷中).

Y.Naito：“拟线性椭圆方程的振荡准则”Nonlineat Anal..（正在出版）。

N.Muramoto: "Existence of self-similar solutions to a parabolic system modelling chemotaxis"Hiroshima Math.J.. (印刷中).

N. Muramoto：“抛物线系统建模趋化性的自相似解的存在”Hiroshima Math.J.（正在出版）。

田中和永: "Uniqueness of positive radial solutions of semilinear elliptic equations in R^N and Sere's non-degeneracy condition" Communications in Partial Differential Equations. (発表予定).

Kazunaga Tanaka：“R^N 和 Sere 的非简并条件下半线性椭圆方程的正径向解的唯一性”偏微分方程中的通信（待提交）。