临界点理论，隐函数定理方法及其应用

On this projection we plan to study some new problems of nonlinear analysis, including: (1) study Poincare-Hopf theorem and perturbation theorem transforming degenerate critical point set into nodegenerate critical point set (involving the extension of Marino-Prodi perturbation theorem), and make it fit for the better application of critical theory. (2) develop a general form of implicit function theorem, splitting theorem and bifurcation theory, especially, make our results applied to nonlinear problems. (3) study Fucik spectrum and the computation of critical groups of critical points of homogeneous functional and also jumping nonlinear problems. (4) give the estimates of the best constants of Poincare inequality on the ground state manifolds with magnetic vector fields. These theories and their applications are confronted with many challenging problems. For example, basic properties of Fucik spectrum curves with high-dimensional domains are hotspot issues the people conerned for a long time. We hope to make research on these problems by introducing new ideas and new approaches. The study and the solvement of these problems not only promote the development of critical point theory, but also supply new instruments for investigating qualitatively and quatitatively on the existence,multiplicity and sign-changing of solutions of nonlinear equations.

本项目拟研究非线性分析中一些新问题，具体研究包括：（1）研究Poincare-Hopf定理以及从退化临界点集到非退化临界点集的扰动定理（包括Marino-Prodi扰动定理的更一般形式），使得临界点理论有更好的应用。（2）发展更一般形式的隐函数定理，分裂定理及分支理论。特别是将我们已建立的相关结果应用于非线性问题。（3）研究Fucik 谱及相应泛函的临界点临界群计算和跳跃非线性问题。（4）研究基态流形上带磁场的Poincare不等式。这些理论及应用面临许多具有挑战性的问题如区域高维情况下的 Fucik 谱线的基本性质的研究是长期以来人们关注的热点问题，我们希望利用一些新的思想和方法来研究这些问题。这些基本问题的研究和解决不仅会推动临界点理论本身的发展，而且从定性和定量两方面为研究非线性方程解的存在性，多重性以及变号性提供新的工具.

本项目发现一种新的更一般形式的同调环绕，这种环绕本质上利用了Nehari流形的结构, Benci-Rabinowitz广义鞍点方法只能得到一个环绕结构，而我们的方法在理论上能提供多个环绕结构。更重要的是用同调论我们可以区分得到的临界点，因此这一方法为寻求多解提供了新途径。发表了一篇近百页的长文，率先系统地讨论了Schrödinger算子的Fučík谱线和第四个解的存在性并建立了在R^n中的弱极大值原理，紧嵌入的缺失和本质谱的出现使得解决这一问题十分困难复杂。