非合作椭圆系统与强不定薛定谔方程的可解性

This project will use variational methods to investigate asymptotical linear noncooperation elliptic systems and the Schr?dinger equation with periodic potentials, and establish existence of solutions for resonant problem, and discusses the conditions under which there exist multiple solutions in case of near resonance. . Noncooperation elliptic systems and the Schr?dinger equation are very important models in Mathematical physics, also are two types of strongly indefinite problems of great concern in nonlinear analysis fields. In recent two decades, many domestic and foreign scholars have used nonlinear functional analysis methods to study the existence and multiplicity of weak solutions, and obtained rich and deep results, but existing results less consider the existence of solutions in case of resonance (special the existence of nontrivial solutions) as well as multiplicity of solutions in case of near resonance. This project will use existing methods for strongly indefinite problems to establish results in this two aspects, and summarize mutual characteristics and intrinsic difficulty for the two types of strongly indefinite problems through the analysis of specific problems, and find new ideas in study of strongly indefinite problems, aims to develop and enrich variational method theory, and at the same time enhance people''s understanding and in-depth knowledge of the two types of important models in Mathematical physics.

本项目拟运用变分方法研究渐近线性的非合作椭圆系统和周期位势Schr?dinger方程，将建立共振问题解的存在性，并探讨接近共振情形下的多解条件。. 非合作椭圆系统与Schr?dinger方程是数学物理中的重要模型，也是非线性分析研究领域十分关注的两类强不定问题。近二十年，众多国内外学者运用非线性泛函分析方法研究其弱解的存在性与多重性，已取得了丰富而深刻的结果，但现有结果较少涉及共振情形的可解性（特别是非平凡解的存在性）以及接近共振情形下的多解性。本项目将运用现有处理强不定问题的方法建立这两方面的结果，并通过分析具体问题，总结这两类强不定问题的共有特点及其研究中的本质困难，探寻研究强不定问题的新思路，旨在发展和丰富变分法理论，同时加深人们对数学物理中的这两类重要模型的认识和理解。

本项目运用变分方法研究了椭圆系统解的存在性和多重性。具体而言，研究有界区域上一类渐近线性的非合作椭圆系统，利用广义的Landesman-Lazer型条件、Benci和Rabinowitz提出的无穷维环绕定理建立了共振问题的可解性，并结合利用局部鞍点定理证明了近共振问题的多解性；研究有界区域上一类半线性椭圆方程，利用经典的局部环绕思想建立了局部超线性问题解的存在性。