具有临界指数增长的椭圆型方程若干问题的研究

项目名称： 具有临界指数增长的椭圆型方程若干问题的研究

项目编号： No.11201186

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王俊

作者单位： 江苏大学

项目金额： 23万元

中文摘要： 本项目主要利用变分方法研究具有临界指数增长的椭圆型方程及其相关问题。这些问题与力学，物理，天文和材料科学等学科有密切联系，具有重要的应用背景和理论价值。但由于缺乏紧性条件，这些问题的研究有很大难度，本项目力求突破这一难点。首先，研究具有临界指数增长的Schr?dinger方程、p-Laplace方程、Schr?dinger-Possion系统、Kirchhoff方程和拟线性椭圆方程中解的存在性及其相关问题，重点关注半经典情况下解的存在性及其性质，这同样是物理学家关注的重点。然后再对具有临界指数增长的强不定椭圆型方程和Dirac方程中解的存在性和多重性等问题进行深入探讨，得到一些新的研究成果，特别关注在凸凹组合的非线性增长条件下系统解的存在性及其性质。

中文关键词： 变分方法；临界点；椭圆方程；基态解；

英文摘要： In this project we mainly study some problems on elliptic equations with critical growth by variational methods. Such problems are related to Mechanics, Physics, Astronomy and Material science etc, which have great meanings both in theory and practice. However, because of the lack of compact condition, it is difficult to solve these problems. We shall do our best to overcome this difficulty in this project. At first, we study the existence of solutions for Schr?dinger equation, p-laplace equation, Schr?dinger-Possion system, Kirchhoff equation and qusilinear elliptic equation with critical growth. Moreover, we are also interested in the existence of solutions for semiclasscal case, which is also a focus for physicists. Then, we study the existence and multiplicity of solutions for strongly indefinite elliptic equations and Dirac equations. In particular, we also care about the existence of solutions for strongly indefinite problems with convex and concave nonlinearities.

英文关键词： Variational Method；Critical points；Elliptic equations；Ground state solutions；