半线性椭圆方程的边界爆破解

Elliptic boundary blow-up problems, which arise naturally when studying models from such areas as Riemannian geometry, mathematical physics, mathematical biology, have attracted a great deal of researchers’s interests, such as Nirenberg, Brezis and so on. Existences and uniqueness is the main difficulty of this problems for singularity of solutions near boundary, therefore, understanding of blow-up phenomenon is a vitally important and challenging questions. Existence, uniqueness and asymptotic behavior of boundary blow-up solutions to semilinear elliptic equations will be studied in this project by Karamata regular variation theory and constructing super-sub solution. More precisely, firstly, we will introduce a new class functions to describe the vanishing rate of weight function and deal with the competition between the vanishing rate of weight function and the growth rate of nonlinear term of elliptic equations, which is an open problem and a very subtle question. Secondly, we will establish a unified and explicit asymptotic formula of boundary blow-up solutions to elliptic equations when nonlinear term belongs to regular variation functions or Γ-varying functions, unlike earlier works, boundary behavior of boundary blow-up solutions was dealt separately for different rate of nonlinear term at infinity and the asymptotic behavior of boundary blow-up solutions was described as the solution to a one-dimensional problem or an integral equation. Finally, we will investigate the existence, uniqueness and asymptotic behavior of boundary blow-up solutions to elliptic systems provided the nonlinear terms belong to regular variation functions or Γ-varying functions for the first time, improving the known results for special nonlinear terms.

源于黎曼几何、数学物理、生物数学等诸多领域的椭圆边界爆破问题，因其解的奇异性使得研究解的存在唯一性非常困难，吸引了大量研究人员如Nirenberg、Brezis等人的兴趣。边界爆破解的研究是一项重要且具挑战性的课题，本项目将通过构造上下解，应用Karamata正规变化理论研究半线性椭圆方程边界爆破解的存在唯一性和渐近行为。具体来说，首先，引入新的函数刻画权函数的衰减速率，进而研究权函数的衰减速率与非线性项增长速率之间的竞争，该问题是一个非常棘手的公开问题；其次，建立当非线性项分别属于正规变化函数和Γ变化函数的统一处理模式而非根据非线性项不同的增长速率分开处理，并建立解的边界渐近行为的显式刻画而不是通过某个常微分方程或者积分方程的解刻画；最后，研究当非线性项分别属于正规变化函数或Γ变化函数时椭圆方程组边界爆破解的存在唯一性和渐近行为，拓展已有的特殊非线性项情形的相关结果。

该项目主要研究椭圆型偏微分方程解的相关性质，一方面研究椭圆边界爆破解的渐近行为，运用Karamata变化理论，结合上下解方法，建立了无论非线性项是快速变化还是正规变化函数时，边界爆破解的渐近行为的统一处理方式，同时也考虑了区域边界的平均曲率对边界爆破解的二阶渐近行为的影响。另一方面，运用局部化方法，构造适当的上下解，建立了当权函数的边界行为与边界点的位置有关时，边界爆破解的唯一性和渐近行为。最后，考虑了非强制拟线性椭圆方程解稳定性，讨论了低阶项的正则化效应，当非线性项指数足够大时，对应方程的解在极限意义下具有稳定性。另外，该项目也同时研究有关抛物方程解的相关性质。