椭圆型偏微分方程中若干奇异问题的研究

In this project, we will study some singular problems in elliptic equations by using nonlinear functional analysis (order method, topological degree theory, variational method and so on), theory and techniques of elliptic equations. Specifically, we will focus on the following problems. The first problem concerns the isolated singularity and classification of solutions of elliptic equations (the origin belongs to the domain). For the strongly singular potential and weakly singular potential, respectively, the nonexistence, existence, local properties, complete classification of positive solutions will be considered. Secondly, we will consider elliptic equations with Hardy potential, which is generated by the distance function from the boundary. Here, the existence of dead core solutions and positive solutions, branch of positive solutions, classification of positive solutions will be studied. And we will also consider the existence, uniqueness and symmetry of positive solution in half-space. Our third problem is concerned with boundary blow-up problems. We will investigate the existence, multiplicity and uniqueness of boundary blow-up solutions, optimize some sufficient conditions of uniqueness, establish sufficient and necessary condition with respect to the existence of strictly convex solutions, and consider analytical properties of boundary blow-up solutions. Finally, we are interested in elliptic equations with isolated singularity on the boundary. We will study the existence, nonexistence, and the classification of positive solutions. We will establish the exact behavior at the singular point of positive solutions with the non-removable singularity. This project can contribute to the development of theory and approaches of nonlinear functional analysis and elliptic equations.

本项目拟利用非线性泛函分析（序方法、拓扑度理论、变分方法等）、椭圆型方程的相关理论和技巧研究椭圆型方程中几个奇异性问题，具体包括：（1）椭圆型方程中的孤点奇异性和解的分类问题(原点在区域内)，分别对强奇异位势和弱奇异位势的情形研究正解的不存在性、存在性、解的局部性质、解的完整分类等；（2）含有到边界距离产生的Hardy位势的椭圆型方程，去研究死核解的存在性、正解的存在性、正解的分支和正解的分类，同时研究半空间上正解的存在唯一性和对称性；（3）椭圆型方程的边界爆破问题，研究边界爆破解的存在性、多解性、唯一性等，将优化唯一性的充分条件，给出严格凸解存在的充分必要条件，以及研究解的解析性质；（4）椭圆型方程在区域边界上的孤点奇异问题，研究正解的存在性、不存在性、正解的分类，并给出不可去奇性的解在奇异点处的准确性质。本项目的研究将对非线性泛函分析与椭圆型偏微分方程理论的发展起到一定的促进作用。

奇异问题是椭圆型方程中的热点问题，其相关的研究有着很好的理论意义。项目通过利用非线性泛函分析理论中的上下解方法、变分方法以及偏微分方程的相关理论技巧，研究如下几个问题：（1）研究带强奇异位势和Hardy位势（由到区域边界的距离生成）的椭圆型方程，并就不同的参数讨论正解的存在性、不存在性，同时建立正解的边界爆破速率；（2）研究带奇异位势（由到原点的距离生成）的椭圆型方程，并建立当该位势是强奇异时正解的存在性、不存在性以及正解在原点处的爆破速率等结论，同时对弱奇异位势的情形，给出正解的一些分类；（3）研究带奇异权函数的椭圆型方程，就非线性项是超线性和次线性情形，分别给出正解的先验估计，同时建立正解的不存在性、存在性和唯一性等结果；运用Nehari流形方法，研究带奇异权函数的椭圆型方程组，给出非平凡正解的存在性；（4）研究带权函数的非局部椭圆型问题，借助变分法和分支理论，给出正解的存在性和正解的分支结构。主要研究成果发表在J. Diffeerential Equations、 Proc. Roy. Soc. Edinburgh Sect. A 、Differential Integral Equations等数学专业杂志上，研究结果使我们对部分椭圆型奇异问题有更深刻的理解和认识，研究方法和结论有具体的理论意义。