Mathematical Sciences: Existence and Nonexistence of Positive Solutions for Some Superlinear Problems

本课题研究了一类超线性椭圆型边值问题正解的存在性和不存在性；典型的情况是非线性是多项式型的。特别强调将放在领域，如球，环空，穿孔球，和外部领域。主要通过不动点定理和次理论来讨论正解的存在性。不存在性问题将主要通过振荡理论和Pohozaev恒等式来研究。提出的问题是由气体动力学、非线性热生成、量子场论和黎曼几何等各种情况引起的。这个项目的一个组成部分将是一些课程和研讨会，使本科生和硕士研究生熟悉当前的数学研究。

The purpose is this project is to study the existence and nonexistence of positive solutions to superlinear elliptic boundary value problems; the typical case is where the nonlinearity is of polynomial type. Particular emphasis will be placed on domains such as balls, annuli, punctured balls, and exterior domains. Existence of positive solutions will be mainly addressed though fixed point theorems and degree theory. Nonexistence problems will be mainly studied through oscillation theory and Pohozaev identities. The proposed problems are motivated by a variety of situations arising in gas dynamics, nonlinear heat generation, quantum field theory, and Riemannian geometry among others. An integral part of this project will be a number of courses and seminars leading to familiarizing undergraduate and masters degree students with current mathematical research.