拟线性椭圆型方程解的存在性与集中性

Quasi-linear elliptic equations are an important class of partial differential equations, which have affluent physical backgrounds. They often appear in many nonlinear physical models, such as reaction-diffusion theory, non-fluid mechanics, non-Newtonian filtration and the gas flow in porous mediums. Based on the existing literature, applying variational methods and the critical point theory, this project will focus on the following core issues of the quasi-linear equations: the existence and multiplicity of the nontrivial solutions, the ground state solutions of Nehari type and the semiclassical solutions, the concentration, decay and regularity of the solutions. By deepening the mathematical tools, we will extend and develop the nonlinear analysis methods and techniques to obtain a number of new and essential results, and promote the development of the qualitative theory of quasi-linear elliptic equations.

拟线性椭圆方程是一类重要的偏微分方程，具有丰富的物理背景，出现在许多非线性物理模型中，例如反应扩散理论、非流体力学、非牛顿过滤和多孔介质中的气体流量等。本项目将借助变分方法与临界点理论，在已有文献的基础上，重点研究拟线性椭圆方程的核心问题：非平凡解、Nehari型基态解和半经典解的存在性与多重性，解的集中性、衰减性和正则性等。发展和开拓非线性分析方法、技巧，深化数学工具，对所研究的问题获得若干全新的、本质性的结果，推进拟线性椭圆方程定性理论的发展。

拟线性椭圆方程是一类重要的偏微分方程，具有丰富的物理背景，出现在许多非线性物理模型中，例如反应扩散理论、非流体力学、非牛顿过滤和多孔介质中的气体流.量等。本项目借助变分方法与临界点理论，在已有文献的基础上，重点研究拟线性椭圆方程的核心问题：非平凡解、Nehari 型基态解和半经典解的存在性与多重性，解的集中性、衰减性和正则性等，获得了若干全新的、本质性的结果，发展和开拓非线性分析方法和技巧，推进了拟线性椭圆方程定性理论的发展。