具有低阶项的椭圆方程解的研究

Elliptic equation is an important partial differential equation. It is widely used in problems of mathematics, physics and engineering. Based on Karamata regular variation theory, this project will be devoted to the research on the solutions for two kinds of elliptic equations with low order terms. Firstly, the existence，uniqueness and boundary asymptotic behavior of blow-up solutions for p-Laplacian elliptic equations will be studied. So far, research on the blow-up solutions of this kind of equations has not been enough. The result of the second order expansion of blow-up solutions near the boundary is even less. The project is proposed to prove the existence of the blow-up solutions by means of perturbation、sub-supersolution method and prior estimates and so on. Moreover, by Karamata regular variation theory, the boundary asymptotic behavior and uniqueness of blow-up solutions will be studied. And, much attention would be paid on the second order expansion of blow-up solutions. Secondly, regularity of solutions of the Dirichlet problem for second order elliptic equation will be studied. By means of Karamata regular variation theory, the project plans to introduce a new weighted generalized Morrey space, discuss the real variable theorey of this function space, and then study the regularity of the solutions for the elliptic equations. The study of this project will enrich the results of elliptic equations and the harmonic analysis, and provide theoretical basis for other subjects.

椭圆型方程是一类重要的偏微分方程，它在数学、物理和工程技术中有着广泛的应用。本项目致力于利用Karamata正规变化理论研究两类具有低阶项的椭圆方程的解。 其一是p-Laplacian椭圆方程爆炸解的存在性、唯一性和边界渐近行为。到目前为止，关于此类方程爆炸解的研究尚不够充分，有关爆炸解在区域边界附近二次展式的结果更是偏少。本项目拟借助摄动、上下解、先验估计等方法证明爆炸解的存在性；并基于Karamata正规变化理论，研究爆炸解的边界渐近行为和唯一性，重点关注解的二次展式。其二是具有VMO系数的二阶椭圆方程Dirichlet边值问题解的正则性。本项目拟借助于Karamata 正规变化理论引进一类新的加权广义Morrey空间，讨论此函数空间的实变理论，而后对椭圆方程解的正则性进行研究。本项目的研究必将丰富椭圆型方程和调和分析中的有关结果，并为其他学科相关问题的研究提供理论依据。

具有低阶项的椭圆方程来源于多个应用学科，关于其解的存在性和解的性质的研究具有重要的科学价值和现实意义。本项目重点研究了非线性椭圆方程奇异边值问题解的存在性、解的边界渐近行为和其它一些相关问题，主要研究成果包括：(1)研究了一类半线性椭圆方程Dirichlet边值问题的解在有界区域边界附近的二阶展式；(2)研究了半直线上一类二阶半线性椭圆方程边值问题的正解在0点附近的爆破速率；(3) 证明了一类变系数p-Laplacian椭圆方程Dirichlet边值问题解的存在性，并刻画了解在区域边界附近的精确渐近行为；(4)刻画了一类变系数无穷拉普拉斯方程Dirichlet零边值问题的解在有界区域边界附近的渐近行为；(5)刻画了一类变系数无穷拉普拉斯方程边界爆破问题的解在有界区域边界附近的渐近行为；(6)证明了一类半线性椭圆方程在有界区域上边界爆破问题解的存在性；(7) 研究了其它一些与本项目相关的内容：调和分析中一些重要算子的估计以及一类非线性方程解的正则性等。