p-Laplacian特征值相关问题的研究是当代非线性科学研究的一个热点课题。从上世纪七十年代开始它一直受到国内外众多知名数学家的广泛关注，并取得了很多有价值的研究结果。然而，迄今为止其特征值相关的一些问题还没有完全研究清楚。本项目旨在利用拓扑度理论研究p-Laplace算子在Dirichlet边值条件下所有特征值的存在与分布，并在此基础之上，利用临界点理论、山路引理、上下解方法以及非线性分析技巧，研究在更一般条件下p-Laplacian方程跨共振问题解的存在性与多重性。探讨和总结不同方法建立的特征值之间内在联系和本质差别，希望从中找到研究非线性微分算子谱的一种新方法。我们的研究将会为人们进一步研究p-Laplacian相关问题提供重要的理论依据。

The related problems of p-Laplacian eigenvalue has been one of the hottest issues in recent nonlinear scientific research. Since the 1970s it has received extensive attention of domestic and international numerous well-known mathematician, and a lot of valuable results are obtained. However, so far some eigenvalue related problems have not been fully investigated.The aim of this project is to study the existence and distribution of all eigenvalues of the Dirichilet problem for p-Laplacian by using the theory of topological degree, and the existence and multiplicity of solutions of across resonance problems for p-Laplacian equations by critical point theory, mountain pass lemma, upper and lower solution method and nonlinear analysis techniques, beside, to explore and summarize the internal connection and essential differences beween eigenvalues established by different methods, hoping to find a new method to study the spectrum of nonlinear differential operator. These researches will provide an important theoretical basis for further research on related problems of p-Laplacian.