非线性椭圆型方程解及其性态的研究

This project is to study some nonlinear Schr?dinger equations and coupled nonlinear Schr?dinger system arising from mathematical physics by using variational method and critical theory. The problems include: 1) Ambrosetti open problem; 2) the existence of nontrivial solutions and infinitely many non-radial solutions (nodal solutions) for the quasilinear Schr?dinger equations with supercritical growth nonlinear term; 3) the exsitence and concentration of nodal solutions、the existence and concentration around degenerate critical points of the potential in the whole space for coupled nonlinear rSchr?dinger system. These problems have closed relations with the research of the classical mechanics, quantum field theory, nonlinear optical, etc, and is the international hot topic. To solve these problems, we need several subjects, such as differential equation, functional analysis, algebra and topology, etc. The completion of these problems not only develop a new method, reveals the new laws, but also has an important academic value and broad application prospect.

本项目拟利用变分法及临界点理论研究数学物理中某些非线性Schr?dinger 方程及耦合非线性Schr?dinger系统。所涉及到的问题包括：1）Ambrosetti的公开问题；2）拟线性Schr?dinger 方程当非线性项超临界增长时非平凡解存在性、无穷多非径向解（变号解）的存在性；3） 耦合非线性Schr?dinger系统变号解的存在性及其集中性态、全空间中非平凡解的存在性及其在位势某些退化点附近的集中性。这些基问题与经典力学、量子场理论、非线性光学等研究有密切的关系，是目前国际上的热门课题，解决这些问题需要涉及到微分方程、泛函分析、代数与拓扑、几何等多个学科。这些问题的解决不仅能发展出新的方法，揭示出新的规律，而且具有重要的学术价值和广泛的应用前景。

本项目拟利用变分法及临界点理论研究数学物理中某些拟线性Schrödinger 方程及耦合非线性Schrödinger系统, 这些基问题与经典力学、量子场理论、非线性光学等研究有密切的关系，是目前国际上的热门课题，解决这些问题需要涉及到微分方程、泛函分析、代数与拓扑、几何等多个学科。这些问题的解决不仅能发展出新的方法，揭示出新的规律，而且具有重要的学术价值和广泛的应用前景。我们得到了几类含次临界指数或临界指数的非线性Schrödinger 方程非平凡解的存在性。