Studies on Eigenvalue Problems of Nonlinear Elliptic Equations

非线性椭圆方程特征值问题的研究

基本信息

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

项目摘要

(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：研究半参数椭圆方程的特征值问题。我们建立了（变化）特征值和特征功能的渐近特性。还研究了两参数Ambrosetti-Prodi问题。我们研究了参数与溶液数量之间的关系。（2）椭圆方程的正解：在无界域中考虑了半连续的二阶椭圆形效率。（3）准线性普通微分方程的阳性解：准线性的普通微分方程的阳性解决方案的阳性解决方案的阳性解决方案，其主要术语是一定术语是一二二曲 - 拉普拉曲霉。我们获得了阳性溶液的异步表示。作为这些结果的应用，我们显示了用于准线性椭圆方程的外部差异问题的几种类型的正溶液。（4）描述霉菌聚集现象的数学模型：我们考虑由Keller和Segel引入的抛物面系统的自相似溶液来描述用于化学型模型的聚合现象。我们阐明了参数与自相似解决方案的数量之间的关系。（5）准线性椭圆方程和椭圆系统的非负非平地解决方案：我们为准椭圆形方程式建立了必要和/或足够的条件，以实现quasilineareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareareare的条件。还获得了几种liouville型定理。