来自几何及物理学中的几类高阶非线性椭圆型方程解的研究

Recently, high order nonlinear partial differential equations have been paid more and more attention. This type of equation has a deep and definite application background, such as geometry, physics and mechanical engineering. On the other hand, during the mathematical study of such equations, various challenging problems have been put forward, and some new mathematical phenomena have emerged. In this proposal, we intend to study the polyharmonic partial differential equations with exponential nonlinearity，we will investigate the existence of stable solutions、the stability of entire solutions、finite Morse index solutions ect about the classification of the solutions; Also, we intent to study the classification of the fourth order elliptic equations with sobolev critical exponent , investigate the global precise behavior-periodicity, local asymptotic symmetry ect, we also intend to study the connection between qualitative properties of solutions and the geometrical and topological properties of the underlying domains.

本项目将系统地研究高阶非线性偏微分方程。此类方程具有深刻明确的几何、物理、机械工程等应用背景；在研究过程中，对数学也提出了许多挑战性问题，并且出现了一些新数学现象。特别地，重点研究具有指数非线性项的多重调和方程，探索在全空间上整体稳定解的存在性、整体解的稳定（不稳定）性，有限Morse指标解等分类结果；研究具有Sobolev 临界指标的四阶椭圆方程在原点奇异的解的分类，重点研究方程解精确的整体行为-周期性、局部渐近对称性等，探索区域的几何与拓扑性质对解集的影响。

本项目将系统地研究高阶非线性偏微分方程。此类方程具有深刻明确的几何、物理、机械工程等应用背景；在研究过程中，对数学也提出了许多挑战性问题，并且出现了一些新数学现象。特别地，重点研究具有指数非线性项的多重调和方程，探索在全空间上整体稳定解的存在性、整体解的稳定（不稳定）性，有限Morse指标解等分类结果；研究具有Sobolev 临界指标的四阶椭圆方程在原点奇异的解的分类，重点研究方程解精确的整体行为-周期性、局部渐近对称性等，探索区域的几何与拓扑性质对解集的影响。