本项目拟揭示相对场算子、曲率型算子及p-Laplace算子等非线性常微分算子的"谱"结构，建立其相应非线性问题的全局分歧定理，进而获得带各种边界条件和各种非线性项的相对场方程、曲率型方程及Monge-Ampere方程解的存在性、唯一性、多解性及解的全局分歧结构。拟解决或至少部分解决Hakl-Torres 关于不定权周期边值问题、Torres关于相对场方程、Lopez-Gomez关于曲率型方程等三个公开问题。研究结果不仅会解决数学领域内的一些大问题，而且对解决其它科学领域中的非线性问题可望提供重要的理论指导。

The proposal will aim to reveal the spectral structures of relativistic operator, mean curvature operator as well as p-Laplace operator, establish global bifurcation theorems for the corresponding nonlinear problems, and further obtain the existence, uniqueness, multiplicity and global bifurcation structure of solutions for relativistic equation, mean curvature equation and Monge-Ampere equation with several kinds of boundary conditions and nonlinearities. Moreover, we shall solve or partly solve the open problems of Hakl-Torres about periodic boundary value problems with indefinite weights, Torres about relativistic equation, and Lopez-Gomez about mean curvature equation. The research is expected not only to solve some important mathematical problems, but also to provide theoretical implications for computing the numerical solutions of nonlinear problems arising in other scientific fields.