Multiple existence and structure of solutions for semilinear elliptic equations.

半线性椭圆方程解的多重存在性和结构。

基本信息

批准号：
16540179
负责人：
KAJIKIYA Ryuji
金额：
$ 2.37万
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：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540179/
关键词：
elliptic equation variational method multiple solutions sublinear ellintic equation p-Laplace equation bifurcation of solution 特異係数関数無限大ラプラシアン放物型方程式解の漸近挙動半線形楕円型方程式比較定理 variational method symmetric mountain pass lemma 偶汎関数

项目摘要

We discover a new critical point theorem related to the symmetric mountain pass lemma. Our theorem asserts that an even functional on a Banach space has a sequence of critical points converging to zero. By applying it to a sublinear elliptic equation, we prove the existence of infinitely many solutions under a very weak condition.When the, nonlinear term is not odd, we prove that a sublinear elliptic equation has infinitely many solutions. By considering the Lagrangian functional associated with the elliptic equation as a perturbation from an even functional, we use the symmetry of the functional to prove the existence of solutions. 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 except for our results.We prove the regularity of solutions for one-dimensional p-Laplace equations with singular coefficients. By using it, we give a necessary and sufficient condition for the existence of eigenvalues. Then we investigate the structure of the bifurcation of solutions for the one-dimensional p-Laplace equations. Moreover, by using the number of zeros of solutions, we study the direction of the bifurcation branch and show the global existence of the bifurcation curve.

我们发现了一个与对称山通行证有关的新临界定理。我们的定理断言，在Banach空间上的均匀功能具有一系列临界点的序列，趋于零。通过将其应用于均匀椭圆方程式，我们证明了在非常弱的条件下存在无限的许多解决方案。当非线性术语不奇怪时，我们证明了sublineareareareare elliptic equariptic equariptation无限的解决方案。通过将与椭圆方程相关的拉格朗日功能视为与均匀功能的扰动，我们使用功能的对称性来证明解决方案的存在。已经研究了超线性椭圆方程的多个解决方案的存在。然而，除了我们的结果外，对于均方根椭圆方程的多个溶液知之甚少。我们证明，具有单数系数的一维p-laplace方程的解决方案的规律性。通过使用它，我们为特征值的存在提供了必要和充分的条件。然后，我们研究了一维p-Laplace方程的溶液分叉的结构。此外，通过使用溶液的零数，我们研究了分叉分支的方向，并显示了分叉曲线的全球存在。