Research on the existence of solutions of differential equations and their properties via functional analysis methods

泛函分析方法研究微分方程解的存在性及其性质

• 批准号: 21540214
• 负责人: SHIOJI Naoki
• 金额: $ 2.91万
• 依托单位: Yokohama National University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009-04-01 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540214/
• 关键词: 関数方程式論; 解析学; 関数解析学; 楕円型方程式; 変分法; 多重解; 正値解; 対称性; 楕円幾何; 双曲幾何; 符号変化解; 解の多重性; Sobolevの臨界指数; 変動指数

For elliptic problems which have Nehari manifold structures, there were some results on the numbers of their positive solutions. However, the numbers of sign-changing solutions were not studied well. By using category on the domain, we give a result on a number of sign-changing solutions. For elliptic problems which have Sobolev's critical exponent, the existence of solutions whose level are greater that the two times of the level of the limit equation were not discussed well. We gave such an argument to show the existence of the multiple solutions.We also studied the uniqueness of positive radial solutions for elliptic equations. The uniqueness of a positive radial solution of the scalar field equation has a long history. We gave a uniqueness result which is applicable to various equations including the scalar field equation.

对于具有Nehari流形结构的椭圆型问题，关于其正解的个数已有一些结果。然而，符号变化的解决方案的数量没有得到很好的研究。利用定义域上的范畴，给出了一系列变号解的结果。对于具有Sobolev临界指数的椭圆型问题，其水平大于极限方程水平的两倍的解的存在性还没有得到很好的讨论。给出了多解存在性的证明，并研究了椭圆型方程正径向解的唯一性。标量场方程正径向解的唯一性问题有着悠久的历史。我们给出了一个唯一性结果，它适用于包括标量场方程在内的各种方程。

项目成果

UNIQUENESS OF POSITIVE RADIAL SOLUTIONS OF Δu+g(r)u+h(r)u[p]=0 AND ITS APPLICATIONS

• 发表时间: 2014

楕円型方程式Δu+▽ρ▽u/ρ-gu+hu^p=0の正値球対称解の一意性とその非退化性について

论椭圆方程Δu+▽ρ▽u/ρ-gu+hu^p=0正球对称解的唯一性及其非简并性

• 发表时间: 2014
• 作者: P.Baseilhac;T.Kojima;Tatsuo Iguchi;塩路直樹 渡辺宏太郎
• 通讯作者: 塩路直樹 渡辺宏太郎

Partial radial symmetry of Positive solutions for semilinear elliptic equations in a discandits application to the H'enon equation

半线性椭圆方程正解的部分径向对称性在 Henon 方程中的 discandits 应用

• 发表时间: 2012
• 作者: Y. Enatsu;Y. Nakata and Y. Muroya;Yousuke Ohyama;高野恭一;Takeo Kojima;塩路直樹;H. Sumi;大山陽介;塩路直樹
• 通讯作者: 塩路直樹

Two solutions for a Bahri-Coron problem in an annular domain with a thin holc

薄霍尔环形域中 Bahri-Coron 问题的两种解

• 发表时间: 2011
• 作者: 三村尚之;礒島伸;村田実貴生;薩摩順吉;Naoki Shioji
• 通讯作者: Naoki Shioji

Ground state and multiple solutions for a critical exponent problem

关键指数问题的基态和多种解决方案

• DOI: 10.1007/s00030-011-0127-0
• 发表时间: 2012-06
• 期刊: Nonlinear Differential Equations and Applications NoDEA