非对称扰动下的拟线性椭圆方程解的多重性研究

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.

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。

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。本项目致力于运用临界点理论中的扰动方法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问题：（1）在有界域上分别获得了具超二次、次二次以及渐近二次增长的一类反应扩散方程存在无穷多解的结果；（2）在有界区域上研究了带非线性Neumann边值条件的Kirchhoff型方程无穷多解的存在性；（3）在全空间中研究了一类具有一般位势以及超线性次临界增长非线性项的Kirchhoff型方程，获得了方程具有Nehari-Pohozaev型基态解和最小能量解的结果；（4）在全空间中用非Nehari流形方法研究了一类Schrodinger-Poisson系统，在周期以及渐近周期情形下分别获得了基态解的存在性等结果。这些研究结果的获得，促进了椭圆方程理论的发展和应用，具有一定的意义和应用价值。