椭圆方程的Liouville定理与边界奇点

与Caffarelli-Kohn-Nirenberg不等式相关的非线性椭圆问题一直是偏微分方程研究的核心内容之一.有关解的存在性、正则性及奇点的研究已经或正在成为国际数学界的热点.申请人将在与合作者已有工作的基础上,充分利用偏微分方程理论与技巧,综合运用常微分方程、调和分析及非线性分析等多种数学学科,讨论解在无穷远点和孤立边界奇点附近的性质.对无穷远点的情形,本项目拟对解做变换转化为内区域问题,采用爆破分析和双倍引理得到解的分类,解及梯度在无穷远点的爆破速度和奇点的可去性.对具有孤立边界奇点的情形,拟结合移动球面法和反证法研究奇点不可去时定义在上半空间解的对称性和单调性,然后利用开尔文变换和爆破分析获得解及梯度在奇点附近的上界估计,再由移动球面法给出渐近性刻画,最后结合Pohozaev恒等式证明奇点的可去性.这些研究将进一步丰富非线性偏微分方程与分析领域中的空间理论.

The nonlinear elliptic problem related to the Caffarelli–Kohn–Nirenberg inequality has always been one of the core subjects of the partial differential equations. The existence, regularity of solutions and the analysis of singularity has been or is becoming a hot issue in the international mathematics research. Based on the existing work of applicants and collaborators, this subject will be devoted to discuss the properties of the solutions at infinity and near the singularity of these equation with an isolated boundary singularity, via making full use of the theory and techniques of partial differential equations, comprehensively using various mathematical methods and tools such as ordinary differential equation, harmonic analysis and nonlinear analysis. In the case of the singularity at infinity, this project using variable substitution for the solution to transform the problem into an interior domain, and use blow up analysis with double lemma to obtain the classification of the solution, the rate of the solution and the gradient at the infinity and the removability of the singularity. For the case of the isolated singularity at boundary, it is expected to make some progress on the symmetry and monotonicity of the solution when the equation defined in the upper half of the space and the singularity is non removability, using the method of moving spheres and the contradiction method, and then comprehensively use Kelvin transformation and blow up analysis to obtain the upper bound of the solution and the gradient near the singularity. The asymptotic properties for the solution will be further given by the method of moving spheres, and finally combined with the Pohozaev identity to prove the removability of the singularity. These studies will further enrich the foundation theory of nonlinear partial differential equations and the theory of space in the field of analysis.