非对称扰动下的拟线性椭圆方程解的多重性研究

项目名称： 非对称扰动下的拟线性椭圆方程解的多重性研究

项目编号： No.11501190

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 陈静

作者单位： 湖南科技大学

项目金额： 18万元

中文摘要： 拟线性椭圆方程有着广泛而深刻的应用背景，它在物理、生物、化学等多个学科领域中有重要作用，其扰动问题多解存在性的研究是椭圆型偏微分方程领域新颖而又重要的研究课题之一。本项目旨在运用临界点理论中的对称泛函扰动方法，来较为深刻地研究几类带有非对称扰动的拟线性椭圆方程多解的存在性，获得一些不同以往的新结果和新方法。此外，还考虑运用下降流不变集方法和极小极大方法相结合，研究当非对称扰动发生时，一类拟线性椭圆方程变号解的多重性，探索非线性项和扰动项的增长对多解存在性的影响。这些问题的解决或实质性的进展，将促进拟线性椭圆方程理论的发展和应用。

中文关键词： 拟线性椭圆方程；非对称扰动；变号解；边值问题；变分法

英文摘要： Quasilinear elliptic equation has profound background and wide application, it plays an important role in physics, biology, chemistry and other fields. The research on multiplicity of solutions for perturbed quasilinear elliptic equations boundary value problem is a novel and important topic in the field of elliptic partial differential equations. This project aims at studying the existence of infinitely many solutions for several classes of quasilinear elliptic equations with broken symmetry by using perturbative method in critical point theory, and obtain some new results and new methods which are different from ones in some previous papers. Furthermore, by using invariant set of gradient flow, combining with the method of minimax theorems, we shall study the multiplicity of sign-changing solutions for quasilinear elliptic equations perturbed from symmetry，and explore the influence on the growth of nonlinearity term and perturbed term. It will enrich the theory of quasilinear elliptic equations and promote the applications through solving or getting crucial developments on the above problems in our project.

英文关键词： quasilinear elliptic equation;perturbation from symmetry;sign-changing solutions;boundary value problem;variational method