带修正项的Kirchhoff型方程（组）解存在性和集中性的变分方法研究

Kirchhoff-type problems with modified terms are the basic models, which can describe more accurately the complex and changeable non-local phenomena in elastic mechanics and biology mathematics. A remarkable feature of such kind of problems is that they contain both non-local terms and quasi-linear modified terms. In this project, aiming to overcome the double difficulties caused by the interweaving of non-local terms and quasilinear modified terms comprehensively, we will focus on the cases of sign-changing potential, critical frequency potential, decay potential, compactly supported potential, and sub-cubic nonlinearity, and obtain a series of meaningful new results on the existence and concentration of Kirchhoff-type problems with modified terms by applying“double”perturbation method, combining dual transformation method with non-smooth variational method, as well as developing non-Nehari manifold method. Thus, a set of variational methods and techniques for studying the existence and concentration of solutions to such problems are established, which not only inject new contents and create new methods into the development of the theory and application of Kirchhoff-type problems, but also provide new ideas and methods for the study of other non-local problems, such as Schrodinger-Poisson equations and Schrodinger-Choquard equations, and then to promote variational and nonlinear elliptic partial differential equations theory and their applications.

带修正项的Kirchhoff型问题是弹性力学和生物数学中的基本模型，它能更精确地刻画弹性力学和生物数学中复杂多变的非局部现象。该类问题的显著特点是同时含有非局部项和拟线性修正项。本项目拟综合克服非局部项和拟线性修正项交织在一起带来的双重困难，着重关心变号、临界频率、衰减或紧支集位势及次3次非线性项情形，应用“双”扰动方法，结合对偶变换方法和非光滑变分方法，以及发展非Nehari流形方法，获得带修正项的Kirchhoff型问题解的存在性和集中性方面的一系列有意义的新结果。从而，建立一套研究这类问题解的存在性和集中性的变分方法和技巧，不仅为Kirchhoff型问题理论及其应用的发展注入新内容，创造新方法，而且为其它非局部问题，如：薛定谔-泊松方程和薛定谔-肖卡尔方程的研究提供新思想和新方法，进而推动变分理论和非线性椭圆偏微分方程理论及其应用的发展。

带修正项的Kirchhoff型问题是非线性分析领域中的前沿热点问题之一，其研究难点在于如何克服非局部Kirchhoff项或拟线性修正项带来的困难。本项目应用变分方法和临界点理论，主要针对一般的非局部Kirchhoff问题、带有两类拟线性修正项的拟线性薛定谔方程（组）以及奇异扰动的广义拟线性修正项问题非平凡弱解、正解、负解、变号解、规范化解和基态解的存在性、非存在性、多重性以及解的集中性、衰减性和渐近行为开展了一系列研究，获得了一些处理非局部Kirchhoff项或拟线性修正项本质困难的思想、方法和技巧，建立了一套研究该类问题解的存在性和集中性的变分方法和技巧，取得了一些有意义的新研究成果。另外，作为本项目研究获得的思想和方法的启发与应用，我们还研究了几类非线性发展方程解的演化和退化性，并数值模拟了解的相互作用和动力学行为等非线性现象。