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

这个项目的目的是研究 超线性椭圆型方程正解的不存在性 边值问题;典型的情况是， 非线性是多项式型的。 将特别强调 放置在诸如球、环、穿孔球的域上， 外部域。 正解的存在性主要是 通过不动点定理和度理论解决。 不存在问题将主要通过振动来研究 理论和Pohozaev恒等式。 提出的问题是由各种动机， 气体动力学中出现的情况，非线性热生成， 量子场论和黎曼几何等等。 该项目的一个组成部分将是一些课程， 研讨会导致熟悉本科生和硕士 目前的数学研究的学位学生。

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.