含临界指标的非线性Kirchhoff型方程（组）解的存在性和集中现象

This project concerns with the existence of solutions and concentration behavior of solutions to nonlinear Kirchhoff-type elliptic equations (system) involving critical Sobolev exponents. There are many results about the model and its relevant model, but the conclusions for the existence, nonexistence and the properties of ground state solution, and infinitely many high energy solutions are not perfect.. The project is discussed from three following problems: 1) Using the finite reduction method to discuss the existence and concentration behavior of positive solutions for a kind of nonlinear Kirchhoff-type elliptic equations involving critical Sobolev exponents with disappeared potentials and general nonlinear term; 2) Combine the theories of regularity、prior estimates、the concentration compactness and the methods used in 1) to discuss the existence of infinitely many high energy solutions for a Kirchhoff type system with critical Sobolev exponent in the whole space; 3) Apply the methods used above to a quasilinear elliptic Neumann problem in a bounded domain with critical Sobolev exponent, trying to get the existence and nonexistence of infinitely many solutions; 4) By decomposing Nehari manifold, scaling technique and the method of perturbation approximation, we get the existence and concentration behavior of multiple solutions and infinite sign-changing solutions to a kirchhoff equation with singular nonlinear term and critical exponent in whole space.. In summary, the project team selecting the search of this kind has comprehensive physical backgrounds. The significance of this subject is to develop new methods and new tools in nonlinear functional analysis, to promote the development of mathematics branch. It is meaningful.

本项目主要研究临界非线性Kirchhoff型方程(组)解的存在性和集中现象，关于该模型及其相关模型已有许多数值结果，但对于基态解或者无穷多高能量解的（非）存在性及解的相关性质的结论并不完善。. 本项目拟从以下问题进行讨论：1)用有限约化的方法讨论位势消失到无穷且含一般非线性项的临界kirchhoff型方程基态解的存在性和集中现象；2)借助1)的方法结合正则性先验估计和集中紧原理讨论临界kirchhoff型方程组无穷多高能量解的存在性；3)将上述方法应用到临界拟线性椭圆Neumann问题上，得到无穷多解的存在与非存在性；4)利用Nehari流形分解、伸缩技巧和扰动逼近理论得到全空间上含奇异非线性项的临界kirchhoff型方程多解及无穷多变号解的存在性和集中性。. 综上所述，项目组选择该课题研究具有广泛的物理意义，可发展出非线性泛函分析中新的方法和工具，促进数学分支发展，是有意义的。

本项目主要研究了一类含有临界指标的非线性Kirchhoff型方程(组)解的存在性，多解性和集中现象。采用的基本方法是变分法，结合上同调指标的抽象临界点原理证明了临界基尔霍夫p拉普拉斯方程解的存在性和多重性；借助与形变引理和Miranda定理相关的变分法得到了薛定谔基尔霍夫型方程基态变号解的存在性。同时也利用Morse理论，通过计算临界群在原点和无穷远点的衰减性，得到分数次超线性基尔霍夫方程非平凡解的存在性。.本课题的研究具有广泛的物理意义，同时发展出了非线性泛函分析中新的方法和工具，促进了数学分支发展，是有意义的。