变分法是非线性泛函分析的重要理论，也是国际数学研究的热点方向。其主要研究对象是来自数学物理、生物工程等领域的微分方程。申请人与合作者已应用变分法在Hamilton系统、椭圆方程方程(组)等问题的研究上获得了一些结果。本项目拟就位势函数的不同，进一步全面研究非线性Kirchhoff型拟线性Schrodinger方程的解：(1) 在没有Sobolev紧嵌入的条件下，利用局部环绕理论、Morse理论等方法研究(强)不定的Kirchhoff型问题的非平凡解的存在性与多重性；(2) 在一类新的条件下，利用能量比较等技巧研究带有限势阱的超线性Kirchhoff型问题的基态解以及多重解；(3) 研究带周期位势的渐近线性或超线性Kirchhoff型方程基态解以及无穷多几何互异解存在性问题。在此基础上，本项目还将对Schrodinger-Poisson系统以及p-Laplace方程等相关问题进行研究。

Variational method is an important theory of nonlinear functional analysis, and it is also a hot direction of international mathematics research. Its main research objects are differential equations come from the mathematical physics, bioengineering and other fields. Applicant and his collaborators have obtained a series results in the study of Hamiltonian systems, elliptic equations by using variational methods. In this project, we mainly study the Kirchhoff type quasilinear Schrodinger elliptic equations for various potentials without compactness conditions: (1) The existence and multiplicity of nontrivial solutions for (strongly) indefinite Kirchhoff type problems are studied by means of local linking theory and Morse theory in the absence of compact Sobolev embedding. (2) Under some new conditions, the ground states and the multiple solutions of the superlinear Kirchhoff type problem with bounded potential well are studied by means of energy comparison. (3) We study the existence of ground states and the existence of infinitely many geometric distinct solutions for asymptotically linear or superlinear Kirchhoff equations with periodic potentials. On the basis of studying Kirchhoff type problems, this project will also study the related problems in Schrodinger-Poisson systems and p-Laplace equations.