变分法与非线性微分方程

项目名称： 变分法与非线性微分方程

项目编号： No.11471187

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： 毛安民

作者单位： 曲阜师范大学

项目金额： 70万元

中文摘要： 本项目致力于发展临界点理论和拓扑度理论，并用于研究四类失去紧性的非线性变分问题：1)带双参数的薛定谔-泊松系统在全空间上的非径向正解和无穷多个正解；2)参数型非局部基尔霍夫问题解的多重性、变号性质、并揭示参数的实际意义；3)一类椭圆方程在扩张的环形型区域（annular-type）上的径向型（radial-type）正解及渐近性质；4)外部（exterior）区域上椭圆方程的变号解、正解及其局部和整体的分歧结构及最小能量解的渐近行为。研究内容涉及经典传统问题和学科发展当前面临的新问题，涵盖了全空间和临界等情况。采用变分讨论和拓扑方法相结合，临界点和不动点相结合的研究路线，剖析上述问题的解的精细性质，进而揭示干扰元的作用机制。拟采取的研究方案和路线具有众多特色和创新，有众多非线性分析技巧（先验估计、截断、逼近、约化）的应用，研究成果将对非线性分析方法和理论的发展有重要意义。

中文关键词： 变分方法；Morse理论；拓扑度理论；变号解；解的渐近行为

英文摘要： The aim of the project is to study critical points theory and topological degree theory and also consider the following four kinds of nonlinear problems：1) nonlinear Schrodinger-Poisson problem with two parameters in the whole space，we are concerned with the existence of positive non-radial solutions for one parameter small and the other one large；2)we are concerned with existence and non-existence of solutions of some nonlocal Kirchhoff type problems；3) we study the existence of positive radial-type solutions of one semilinear elliptic equation in some annular-type domian with an expanding hole；4) the existence of sign-changing solutions of some kind of nonlinear Schrodinger equation and the local and global bifurcation structure of positive solutions of the system ofnonlinear Schrodinger (or Gross- Pitaevskii)type equations. Such problems arose from semiconductor theory, see the references for more physical background, also comes from biological systems and from the Bose-Einstein condenstates and nonlinear optics. The study involv many methods and technique，such as invariant sets of descent flow，minimax principle，careful analysis of the spectrum，reduction technique，cut-off technique，approximating technique，a-priori bounds of solutions and so on.

英文关键词： Variational method;Morse theory;Topological degree theory;Sign-changing solutions;Asymptotic behavior of solutions