具有临界增长的基尔霍夫型问题的研究

本项目主要针对具有临界增长的变分问题，采用非线性泛函分析的工具和方法对该类问题中的非线性基尔霍夫型（Kirchhoff type）问题孤子型解、多包解的存在性、集中性以及解的性质进行研究，这些方程不但具有强烈的物理意义和应用背景，而且在数学理论上也有重要的意义。目前具有临界增长的基尔霍夫型问题的理论尚不完善，尤其是这类问题孤子型解的存在性和集中性现在还没有任何的结果。因此，急需人们进行发展和创新，基于前人和我们过去的工作，我们将继续开展这类问题的探索和研究。 本项目将致力于解决如下问题：利用畴数理论研究具有位势函数的基尔霍夫型问题孤子型解的集中性和多解性；利用变分方法和临界点理论研究基尔霍夫型问题多包解的存在性和多解性；借助于集中紧性原理研究在临界的情形下基尔霍夫型问题非平凡解的存在性和多解性；研究具有临界增长的变指数的基尔霍夫型问题解的存在性和多解性。

This project primarily have a focus on the research of nonlinear variational problems with critical growth. My research work is devoted to studying the existence, concentration and other properties of soliton solutions and multi-bump solutions of Kirchhoff type equations by using the tool and mothed in nonlinear functional analysis. Those equations have strong physical meaning and practical background. Futhermore, the valuation on the aspect of mathematical thoerem is obvious. Currently, the research of the nonlinear variational problems is far from perfect, especially in the existence and concentration of soliton solutions for Kirchhoff type equation. All of those reasons gives researches the motivation to find some creative ways to solve those kind problems. According the work of other researches and ourselves we will continue to do some researches on those problems. The ultimate goal of this project is to solve these following problems: studing the existence and the energy concentration of soliton solutions of Kirchhoff type equation with potential function by using the Ljustemik-Schnirelman Theory; studying the existence of multi-bump solutions for Kirchhoff type equation with the help of variational method and critial point theory; studying multiplicity of nontrivial solutions for Kirchhoff type equation with critical growth by means of Concentration-Compactness-Principle and varitional method; studying the existence and other properties of the solution of Elliptic equation with variable exponent with the help of variable exponent Concentration-Compactness-Principle.