非线性椭圆方程边值问题的单重特征值的分歧

Bifurcation from Simple Multiplicity of Eigenvalues for Nonlinear Elliptic Equation Boundary Problems

考虑涉及高阶摄动和单重特征值的抽象分歧方程的局部结构 ,在更光滑的假设下 ,得到作为参数的函数小分歧解的精确个数 ;在较弱的光滑性假设及简单奇异点的情况下 ,将分歧定理和 Krasnoselskii- Zabreiko拓扑度定理结合起来 ,得到一个关于对一类函数分歧方程的小解最少个数的存在性结论 .由于这个结论包含局部 L eray- Schauder度的信息 ,因此得到关于一些包含单重特征值的非线性问题多解的有用的条件 .通过先验估计 。

On the basis of considering the local structure of an abstract bifurcation equation involving higher order perturbations and simple eigenvalues and more smooth assumptions, we obtained precise bounds for the number of the distinct small bifurcation of solutions as a function of the parameter. Under less smooth assumptions, combining a bifurcation theorem with a topological degree theorem of Krasnoselskii Zabreiko in the case of a simple singular point, we obtained an existence result on the number of small solutions for a class of functional bifurcation equation. Furthermore, since this result contains the information of local Leray Schauder degree, one obtains a useful condition to study multiple solutions for some nonlinear problems involving simple eigenvalues. We proved several new multiplicity results for nonlinear elliptic boundary value problems via a priori estimates.