非线性偏微分方程的多解和凝聚现象

Much attention has been paid to multiple solutions and concentrations phenomena of nonlinear partial differential equations because of their wide applications. The current proposal focuses on the nonlinear elliptic PDE from Physics and Geometries, such as Lin-Ni-Takagi problem, nonlinear Schrödinger system, singularly perturbed system. By the methods and technologies in PDE, we concern the multiple solutions and phenomena of concentrations, as well as their interior mechanism of interactions, and provide the insight of nonlinear phenomena.

非线性偏微分方程的多解和凝聚现象因其广泛应用，正在受到越来越多的关注. 本项目研究一些源于物理和几何的非线性椭圆偏微分方程，如Lin-Ni-Takagi方程、非线性Schrödinger系统、奇异摄动方程系统等. 主要是用偏微分方程的方法和技巧来刻画这些方程（系统）的多解和所蕴含的凝聚现象, 以及它们内部的作用机理，促进对非线性现象的认识.

本项目主要研究一些源于物理和几何的非线性椭圆偏微分方程。我们得到了Lin-Ni-Takagi方程在流形上解的凝聚现象，非线性Schrödinger系统无穷多正多峰解的存在性和连续非径向对称爆破解的不存在性，和一类奇异摄动方程系统内部尖峰解的存在性和凝聚位置。所用的主要方法是偏微分方程方法和技巧，希望我们的工作能促进对非线性问题的认识。