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.我们在N维欧几合物空间的球或环中研究半线性椭圆方程。令G为正交组的封闭子组。如果在G动作下是不变的，则称为G不变。由于G是正交组的封闭子组，因此它是单位球体上的转换组。事实证明，当且仅当G在单位球上不传递G时，存在G不变的非辐射溶液。2。我们研究淋巴结溶液，该溶液是具有零零的径向对称溶液，对于二阶sublineareareareareareare zeros。我们为每个整数k的k节溶液的存在和唯一性获得了必要的条件。结果意味着，均方椭圆方程的径向对称解由其零数唯一决定。这在组不变解决方案的研究中提供了重要的信息。3.在sublinear椭圆方程中，证明存在无限的许多解决方案，而没有假设非线性项是奇怪的。在这种情况下，与椭圆方程相关的拉格朗日功能甚至都不是，但是它被认为是偶数功能的扰动。已经研究了超线性椭圆方程的多个解决方案的存在。但是，对于均方根椭圆方程的多个溶液知之甚少。