几类典型和退化边界条件下带奇异项椭圆方程解的多重性

This project focuses on the study of the existence and multiplicity of solutions for elliptic equations and systems near resonance with singularity nonlinearity and degenerate and several kind of typical boundary conditions. These equations and systems have fruitful practical backgrounds, and it has important significance in studying the existence and multiplicity of their solutions. Whether or not there exist solutions to these partial differential equations or under what conditions there exist solutions will play a crucial role for those fields which involve nonlinear partial differential equations such as iconography, electromagnetism, optics, quantum mechanics, seismology, and meteorology and so on..We aim at applying critical point theory, variational methods and topological degree theory which are developed recently in methods of nonlinear analysis, and at the same time we take advantage of some analytic techniques of truncated function to deal with elliptic equations and systems near resonance with Dirichlet, Neumann, Robin and even degenerate boundary conditions and with critical exponential terms in order to obtain many more solvable conditions and multiplicity of solutions.

本项目主要研究具有奇异非线性项及几类典型和退化边值条件下椭圆方程和椭圆系统的近共振问题解的存在性和多重性。这些方程和系统有广泛的实际背景，解的存在性和多重性及其性态研究有十分重要意义，这些偏微分方程或系统是否可解或在什么条件下可解，将电磁学、光学、量子力学、地震学、气候学等科技领域所涉及的非线性偏微分方程的研究及计算有很大的促进作用。. 我们的目的是综合运用近年来发展的临界点理论，变分方法，拓扑度等非线性分析方法，结合截断函数等分析技巧对具有奇异非线性项、具有Dirichlet, Neumann, Robin甚至退化等各种边值条件及带临界指数项椭圆方程和椭圆系统的近共振问题进行系统的研究，得到更多更好的可解性条件和多重性结果。

本项目主要研究具有奇异非线性项及几类典型和退化边值条件下椭圆方程和椭圆系统的近共振问题解的存在性和多重性。这些方程和系统有广泛的实际背景，解的存在性和多重性及其性态研究有十分重要意义，这些偏微分方程或系统是否可解或在什么条件下可解，将电磁学、光学、量子力学、地震学、气候学等科技领域所涉及的非线性偏微分方程的研究及计算有很大的促进作用。. 我们综合运用近年来发展的临界点理论，变分方法，拓扑度等非线性分析方法，结合截断函数等分析技巧对具有奇异非线性项、具有Dirichlet, Neumann, Robin甚至退化等各种边值条件及带临界指数项椭圆方程和椭圆系统的近共振问题进行系统的研究，得到了较好的可解性条件和多重性结果，达到了预期课题的目标。