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.我们研究n维欧几里得空间的球或环中的半线性椭圆方程。令G 为正交群的闭子群。如果一个解在 G 作用下保持不变，则该解称为 G 不变量。由于G是正交群的闭子群，因此它是单位球面上的变换群。证明了当且仅当G在单位球面上不具有传递性时，存在G不变的非径向解。2.研究二阶次线性椭圆方程的节点解，它是有零点的径向对称解。我们获得了每个整数 k 的 k 节点解存在且唯一的充要条件。结果意味着次线性椭圆方程的径向对称解由其零个数唯一确定。这为群不变解的研究提供了重要的信息。3.在次线性椭圆方程中，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函数，但它被视为偶函数的扰动。研究了超线性椭圆方程的多重解的存在性。然而，人们对次线性椭圆方程的多重解知之甚少。