复杂非线性椭圆问题研究

项目名称： 复杂非线性椭圆问题研究

项目编号： No.11471147

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 赵培浩

作者单位： 兰州大学

项目金额： 70万元

中文摘要： 随着科学与技术的发展，涌现出一大批复杂非线性问题。复杂非线性椭圆问题是非线性问题的基本问题，也是目前椭圆问题研究的核心，集中体现了非线性问题的难点。本项目拟借助于非线性泛函分析的临界点理论等变分方法以及拓扑度理论、Morse理论等拓扑方法，结合椭圆方程的各种先验估计和调和分析的若干技巧，去研究复杂非线性椭圆问题的可解性、多解性以及解的分析与几何等基本性质。复杂非线性椭圆问题是指具有边界非线性、非局部性、奇异非线性以及非光滑区域上的椭圆问题，椭圆问题主要指p-拉普拉斯问题、变指数椭圆问题以及完全非线性椭圆问题。主要研究在边界上满足某种非线性条件的椭圆问题的变分解以及非变分解,研究非局部椭圆问题以及奇异椭圆问题的可解性与多解性以及解的性质，进一步厘清复杂非线性对椭圆问题的本质影响，为几何、物理等相关问题提供新的思路和方法。

中文关键词： 非线性分析；临界点理论；椭圆方程；存在性；不动点

英文摘要： With the development of science and technology, there emerged a large number of complicated nonlinear problem. Complex nonlinear elliptic problem is a basic problem of nonlinear problem, is also current core of the research of elliptic problem,and embodies a concentrated reflection of nonlinear problem. By using the critical point theories on nonlinear functional analysis such as variational arguments, Morse theory and topological degree theory, and combining with the classical elliptic estimates and the techniques in harmonic analysis, we study the solvablity and the properties of solutions, such as existence, multiplicity and the analytical and geometrical properties of solutions of some elliptic problems with complicated nonlinearties. Here the so-called complicated nonlinear elliptic problem mean the elliptic problem with nonlinearity on the boundary, nonlocal nonlinearity, singular nonlinearity and nonsmooth domain;the elliptic problems include p-Laplacian problem, elliptic problem with variable exponents and fully nonlinear elliptic problems. We are mainly interested in studying the existence of variational solution and non-variational solution of the elliptic problem with nonlinear boundary condition, the solvability and multiplicity of solutions of nonlocal elliptic problem and the problem with singular nonlinearities. Our research will further clarify the complex nonlinear effects on the nature of the elliptic problems, and supply some new ideas and methods to related physical and geometrical problems.

英文关键词： nonlinear analysis;the critical points theory;elliptic equations;existence;fixed points