Existence And Concentration Of Solutions For A Schrodinger Equation

In this article, we use variational methods to discuss the existence and concentration behavior of solutions for the Schrodinger equation. This article mainly consists of two chapters.The first chapter consider the existence of solutions for the Schrodinger equation where 0< r< min{4, N}. Under suitable assumptions on a{x) and b(x), we establish some existence results for this equation by variational methods.The second chapter study a quasilinear Schrodinger equation where V(x) and Q(x) are two continuous real function on RN, ε is a positive parameter. The nonlinearity f(s) is assumed to be of critical exponential growth in the sense of Trudinger-Moser inequality. Under suitable assumptions on the potential functions V(x), Q(x) and the nonlinearity f(s), we are able to es-tablish some existence and concentration results for semiclassical solutions by variational methods, and we observe that the concentration appears at the global maximum point of the nonlinear potential Q(x).