非线性椭圆方程及方程组变号解的存在性及其性质研究

The mathematical model about nonlinear elliptic problems involving singular coefficients and critical Sobolev exponents not only can be used to explain the states between stars, has the profound physical meaning, but also has important value of mathematical theory. It has been one of the key problems of scholars to research. There are many results about the model and its relevant model, but the conclusions for the existence, nonexistence and the properties of infinitely many nodal solutions or high energy solutions are not perfect. By using Concentration Compactness principle, Mountain Pass Lemma etc variational method, we will give in-depth research on the existence and nonexistence of infinitely many nodal solutions, also on the related solution properties for problems which related to astrophysics and the differential geometry. 1) We discuss critical exponents and critical dimensions for nonlinear elliptic problems with singular coefficients, and get the existence and nonexistence of infinitely many nodal solutions. 2) We investigate the infinitely many solutions with large energy level for elliptic systems involving critical exponents through the local Pohozaev identity and perturbation approximation theory. The significance of this subject research is not only to enrich the basic theory of partial differential equation, but also to promote the development of mathematics branch. It is meaningful.

由含奇异系数和临界指标的非线性椭圆问题建立的数学模型，不仅可以用来解释星体间性态，具有深刻的物理意义，而且也具有重要的数学理论价值，一直是学者们研究的重点问题之一。关于该模型及其相关模型已有许多数值结果，但对于无穷多个变号解或者高能量解的存在性、非存在性及解的相关性质的结论并不完善。本项目主要利用集中紧原理、山路引理等变分方法对源于天体物理和微分几何的两个非线性椭圆问题进行深入的研究。1）讨论具有奇异系数非线性椭圆方程的临界维数和临界指数，得出其无穷多个变号解的存在性及非存在性。2）利用局部Pohozaev恒等式和扰动逼近的思想来研究带有临界Sobolev指数的椭圆方程组无穷多个高能量解的存在性及非存在性。该课题研究既能丰富偏微分方程基本理论宝库，又能促进数学分支的发展，从而是有意义的。

本项目主要利用了集中紧原理、山路引理等变分方法对源于天体物理和微分几何的两个非线性椭圆问题进行深入的研究。首先，讨论了具有奇异系数非线性椭圆方程的临界维数和临界指数，得出其无穷多个变号解的存在性及非存在性。项目组用变分方法和常微分方程的打靶法研究了方程解存在时所对应的空间的临界维数，并且得到了区域为单位球时方程变号球对称解的存在性。其次，利用Nehair流形变分方法得到了多调和方程多解的存在性和含有临界指标的方程组多解的存在性。