近年来，许多学者研究了平面上带指数非线性项椭圆型方程解的存在性及解的性质。在本项目中，我们将考虑几类带指数非线性项椭圆型问题解的存在性、以及解的集中现象。主要包括以下三部分：第一部分，在平面上的有界区域考虑一类带有权函数的Dirichlet边值问题，主要研究权函数对解的存在性与集中现象的影响；第二部分，在一类无界的、单连通区域上研究Dirichlet边值问题具有集中现象解的存在性，此部分主要探讨区域的拓扑性质对问题解存在性的影响；第三部分，在平面上的有界区域考虑一类Neumann边值问题，得到解的存在性与集中现象。 我们将主要运用Lyapunov-Schmidt约化方法结合变分方法得到上述结果。

In recent years, the existence and properties of solutions to semilinear elliptic equations with exponential nonlinearity in R^2, have been studied by many researchers. In this project, we will consider the existence and concentration phenomenon of solutions for some semilinear elliptic equations with exponential nonlinearity. There are three parts as follows: In the first part, we consider the Dirichlet boundary value problem with some weight functions in a smooth bounded domain of R^2, the main purpose is to find the relationship between the properties of the weight functions and the existence of solutions. In the second part, we investigate the existence of solutions with concentration phenomenon for Dirichlet problem in an unbounded, simply connected domain in R^2. We focus on studying the topological properties of the domain which effect the existence of solutions. In the third part, we are interested in the existence and concentration phenomenon of solutions for some Neumann boundary value problems in a bounded domain. We will get these results by using Lyapunov-Schmidt method with the variational methods.