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Qualitative theory of solutions for semilinear elliptic partial differential equations

半线性椭圆偏微分方程解的定性理论

基本信息

批准号：
12640197
负责人：
KAJIKIYA Ryuji
金额：
$ 2.18万
依托单位：
Nagasaki Institute of Applied Science
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 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640197/
关键词：
semilinear elliptic equation group invariant solution nodal solution variational method perturbation problem grouop invariant solution nodal solution

项目摘要

1.We study semilinear elliptic equations in a ball or annulus of n-dimensional Euclid space. Let G be a closed subgroup of the orthogonal group. A solution is called G invariant if it is invariant under G action. Since G is a closed subgroup of the orthogonal group, it is a transformation group on the unit sphere. It is proved that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere.2.We study the nodal solution, which is a radially symmetric solution having zeros, for the second order sublinear elliptic equations. We obtain the necessary and sufficient condition for the existence and uniqueness of a k-nodal solution for each integer k. The result means that the radially symmetric solution of a sublinear elliptic equation is uniquely determined by its number of zeros. This gives an important information in the study of group invariant solutions.3.In sublinear elliptic equations, it is proved that there exist infinitely many solutions without the assumption that the nonlinear term is odd. In this case, the Lagrangean functional associated with the elliptic equation is not even, however it is considered as a perturbation from an even functional. The existence of multiple solutions has been studied for the superlinear elliptic equations. However, little is known about the multiple solutions of the sublinear elliptic equations.

1.研究了n维欧几里得空间中球或环上的半线性椭圆型方程。设G是正交群的闭子群。一个解称为G不变的，如果它在G作用下是不变的。因为G是正交群的闭子群，所以它是单位球面上的变换群。证明了存在G不变的非径向解当且仅当G在单位球面上不传递。2.研究了二阶次线性椭圆型方程的节点解，它是一个具有零点的径向对称解。对任意整数k，我们得到了k-节点解存在唯一的充要条件。结果表明，次线性椭圆型方程的径向对称解由其零点个数唯一决定。这为研究群不变解提供了一个重要的信息。3.在次线性椭圆型方程中，证明了在不假设非线性项为奇数的情况下，存在无穷多个解。在这种情况下，与椭圆方程相关的拉格朗日泛函不是偶数，但它被认为是来自偶数泛函的扰动。研究了一类超线性椭圆型方程多解的存在性。然而，人们对次线性椭圆方程的多解知之甚少。