The Existence Of Solutions Of Nonlinear Schr(?)dinger Equation

Nonlinear Schrodinger equation originate from application of applied mathematics,physics and other subjects,it can be used to describe the nonlinear wave physics in various,Such as laser beam propagation in medium with refractive index and amplitude,water wave and plasma wave of ideal fluid on free surface,etc.Based on the practical significance of the nonlinear Schrodinger equation,in this paper,we study the existence of solutions to three kinds of schrodinger equation with different physical phenomena,and the specific contents of this paper are as follows:In Chapter One,we first introduce some research background and research status,and then summarize the structure of this paper and the main research results.In Chapter Two,we mainly give the notation,definitions and formulas used in this paper.In Chapter Three,we study a class of nonlinear equation where ? is bounded region of RN(N?2),?N is N-Laplace operator,f(x,u)is a real function with critical growth.By the variable transformation,a class of nonlinear Schrodinger equations be con-verted to semilinear ones,then we study the conditions of equation,finally,by using Trudinger-Moser inequality,Sobolev embedding inequality and so on,we proved the con-ditions of mountain geometric,by using the Mountain Pass lemma,we proved the exis-tence of nontrivial solutions for this equation.In Chapter Four,we study the following nonlinear Schrodinger equation-?u+V(x)u-?(u2)u=f(x,u),u?H1(RN),N? 3.By the Mountain Pass Lemma,Lions Lemma and Sobolev embedding inequality proves that the existence of soliton solution.In Chapter Five,we consider the following nonlinear Schrodinger equation-?u+V(x)u+?/2[?(u2)]u=f(x,u),x?RN,where V(x):RN?R is a given potential,f(x,u)is real function,? is a constant,N?3.By the variable transformation,Fountain Theorem and Sobolev embedding inequal-ity proves that the equation has multiple solutions.Because variable transformation function is piecewise form,at last we need prove ||L?|| of this solutions is bounded.