Existence And Concentration Of Solutions For Two Classes Of Quasilinear Schr(?)dinger Equations

In this article we prove the existence of positive ground state solutions for two kinds of quasilinear Schrodinger equation, and give some properties about their solu-tions. We consider the following two cases.The first case:We study a quasilinear Schrodinger equation of the form-△u+V(x)u-△(u2)u= h(x, u),x∈Rn, (1) and consider the existence and multiplicity of its solution, where its nonlinearity is periodic asymptotically linear and satisfies a monotonicity condition. We use the gen-eralized Nehari manifold methods to obtain a ground state solution and infinitely many geometrically distinct solutions when the nonlinearity is odd.The second case:we study the existence of nontrivial solutions for the following quasilinear Schrodinger equation-ε2△u+V(x)u-ε2[△(u2)]u= Q(x)h(u), x∈R2, (2) in the whole two-dimension, where V and Q are two continuous real function on R2. The nonlinearity h is allowed to enjoy the critical exponential growth in R2. we are going to use the Trudinger-Moser inequation and mountain pass lemma to study the existence of semiclassical ground state solutions for ε small and describe certain con-centration phenomena of these solutions.This article consists of three chapters. The first chapter is devoted to discuss the introduction including research background and prerequisite knowledge. The second chapter deals with the existence and Multiplicity of the solutions for the equation under the first case. In the last chapter we research the the existence and Concentration of the solutions for the equation about the second case.