全空间中Kirchhoff方程解的存在性问题

Kirchhoff problems are important nonlinear equations with nonlocal terms. The study on the existence of nontrivial solutions for Kirchhoff problems is a very important area in the nonlinear functional analysis theory. The present project mainly studies the following problems: 1) By using the generalized Nehari manifold, we study the existence of ground state solutions for Kirchhoff problems. Under the asymptotically 4-linear growth of non-linearities, we study the existence of solutions for Kirchhoff problems by using the mountain pass theorem and linking theorems. 2) If the potential does not satisfy any compact conditions, we study the existence of infinitely many solutions for Kirchhoff problems with special non-linearities by using Nehari manifold, Morse theory and so on. 3) There are very few researches on the solutions for Kirchhoff problems with critical exponent in high dimension space (dimension larger than three). We study the existence and multiplicity of solutions for these problems by using Nehari manifold and mountain pass theorem, answer some open problems and consummate the existence results on bounded state solutions. By proposing and solving the problems in this project, it will help to deepen the studies of Kirchhoff problems and will extend the application of nonlinear functional analysis in the studies of nonlocal problems.

Kirchhoff方程是一类重要的带非局部项的非线性方程。该方程非平凡解的存在性研究一直是非线性泛函分析理论中一个相当重要的研究领域。本项目主要研究内容包括：1）利用广义Nehari流形方法研究强不定情形下Kirchhoff方程基态解的存在性；利用山路定理和环绕定理等工具，研究非线性项具有渐近四次增长的Kirchhoff方程解的存在性与多重性。2）当位势不满足某种紧性条件时，利用集中紧性原理和Morse理论等工具研究非线性项是特殊函数的Kirchhoff方程无穷多高能量解的存在性。3）关于高维（维数大于3）全空间中带临界指数增长项的Kirchhoff方程解的存在性研究相当少，本项目利用山路定理和Nehari流形等工具研究该类方程解的存在性与多重性，回答部分公开问题并完善束缚态解的存在性。本项目中问题的提出和解决，将有助于深化该方程的研究并拓展非线性泛函分析理论在研究非局部问题中的应用。

本项目研究的Kirchhoff方程是一类带有非局部项的非线性方程，能更好地反映物理学和生物学中某些问题的本质现象。我们利用变分方法和临界点理论研究Kirchhoff型方程非平凡解的存在性与多重性，具体包括：1）研究高维空间中带临界指数增长项的Kirchhoff方程得到了束缚态解的存在性与多重性。2）首先，考虑全空间中带临界增长项的退化Kirchhoff型方程，特别地，我们考虑了高维空间的情形，得到了束缚态解的存在性。其次，研究了一类带非齐次扰动项的离散薛定谔方程，对于位势满足强制条件时得到了该方程解的多重存在性结果。3）首先，研究了全空间中带渐近4-次非线性项的Kirchhoff型方程得到了极小能量变号解的存在性。其次，考虑位势函数快衰减(或为紧支集函数)时带非线性项超4-次增长的Kirchhoff型方程得到了该方程半经典解的存在性与多重性。本项目中问题的提出和解决，将有助于深化该方程的研究并拓展非线性泛函分析理论在研究非局部问题中的应用。