Standing Wave Solutions Of Nonlinear Schrodinger Equations

In this paper,we mainly study the existence of multiple standing wave solu-tions of three kinds of nonlinear Schrodinger equation,one of which is semilinear Schrodinger equation with potential well and the other two of which are quasilinear Schrodinger equations.The thesis consists of four chapters:In Chapter One,we first summarize the research background and current re-search situation of this paper,and then summarize the main work and main results of this paper.In Chapter Two,we study the existence of multiple sign-changing solutions for the following nonlinear Schrodinger equation—?u+V?(x)u=f(x,u),x?RN,where the potential V?(x)has a potential well with bottom independent of the parameter ?>0.We show that as ??? more and more sign-changing solutions of the nonlinear Schrodinger equation exist.The solutions lie in H1(RN)and are localized near the bottom of the potential well.In Chapter Three,we study the existence of multiple positive solutions of the following nonhomogeneous generalized quasilinear Schrodinger equation-div(g2(u)?u)+ g(u)g'(u)|?u|2 + V(x)u= =h(u)+?k(x),x ? RN,where N?3,g:R?R+ is an even differentiable function satisfying lim t?+? g(t)t?-1??>0 for some ??1,h is a continuous nonlinear function covering the case h(t)=|t|p-2t(2<p<?2*),the potential V:RN?R is positive and ?k(x)is a perturbation term with ?>0.Combining the change of variables and variational arguments,we show that the given problem has at least two positive solutions for some ?0>0 and ??(0,?0).In Chapter Four,we study the existence of multiple solutions for a generalized quasilinear Schrodinger equation with indefinite potential-div(g2(u)?u)+ g(u)g'(u)|?u|2 + V(x)u = h(u),x ? RN,where V:RN? R is an indefinite potential function,g:R?R,h:R?R is the appropriate function.By Morse theory,we obtain a nontrivial solution for the problem.In addition,if the nonlinear term is an odd function,we can obtain an unbounded sequence of solutions.