Mathematical Sciences: Some Semilinear Elliptic Problems

摘要 DMS-9622102 Wang 本文研究了一类非线性椭圆型问题的模型方程（包括非线性Neumann问题和半线性Schrodinger方程）解的存在性、定性性质、多解性以及解的分支分支.利用这些模型方程，我们将发展出一些一般的和系统的方法和途径，可用于更一般的问题。该提案将使用变分和拓扑技术，以发展有关问题的分析结果。 偏微分方程是物理世界数学建模的基础。提案中的问题来自应用科学的各个领域。数学分析的作用是提供有关解决方案的定性和定量信息，回答有关唯一性，多样性，形状和增长的问题。

Abstract DMS-9622102 Wang This proposal is concerned with problems of the existence, qualitative properties, and multiplicity of solutions as well as bifurcation branches of solutions for some model equations of nonlinear elliptic problems including a nonlinear Neumann problem and a semilinear Schrodinger equation. Using these model equations we shall develop some general and systematic methods and approaches which can be used for more general problems. The proposal will use variational and topological techniques to develop analytical results for the problems concerned. Partial differential equations form a basis for mathematical modeling of the physical world. The problems in the proposal arise from a variety of areas of applied sciences. The role of mathematical analysis is to provide qualitative and quantitative information about the solutions, answering questions about uniqueness, multiplicity, shape and growth.