Schrödinger方程是量子力学中基本方程，Schrödinger-Poisson方程、Kirchhoff型方程、Bose-Einstein凝聚态方程和拟线性Schrödinger方程在形式上完全是基于Schrödinger方程而推导出来的，这几类方程是常用的数学物理方程，构成了量子力学、弹性力学、流体力学、凝聚态物理的核心基础。许多经典的数学问题都与其密切相关。本项目将借助变分方法与临界点理论，在已有文献的基础上，重点研究上述5类方程的核心问题：驻波解、基态解和半经典解的存在性与多重性，解的集中性、衰减性和正则性等。通过改造一些经典方法，发展和开拓非线性分析新方法、技巧，深化数学工具，对所研究的问题获得若干全新的、本质性的结果，推进非线性椭圆方程定性理论的发展。

Schrödinger equations are fundamental equations in quantum mechanics. Schrödinger -Poisson equations, Kirchhoff-type equations, equations in Bose-Einstein condensates and quasi-linear Schrödinger equations are completely derived from Schrödinger equations in form. These kinds of equations constitute the core of quantum mechanics, elastic mechanics, fluid mechanics and condensed matter physics. Many classical mathematical problems are closely related to those equations. In this proposed project, on the basis of the existing literature, we will use variational methods together with critical point theory to focus on the research of the core issues for nonlinear Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff-type equations, equations in Bose-Einstein condensates and quasi-linear Schrödinger equations. The core issues in detail include the existence and multiplicity of standing waves, ground state solutions and semi-classic solutions, as well as some dynamical properties of the standing waves such as concentration, decay property and regularity, etc. Through reforming some classical methods, exploring and developing new ways/techniques of nonlinear analysis, and intensifying instructional tools, we expect to acquire some entirely new and essential results on those major mathematical problems with significant challenges, and greatly promote the development for the qualitative theory of nonlinear elliptic equations.