Existence And Multiplicity Of Solutions To The Quasilinear Schrodinger Equation

Differential equations are not only the main branch of traditional applied mathematics.but also an important part of contemporary mathematics.Moreover,differential equations are widely used in physics,mechanics and other disciplines.At present,the research of non-linear differential equations,especially nonlinear partial differential equations,has become a trend.As an important class of nonlinear partial differential equations,the existence of quasi-linear Schrodinger equation is always of great interest to scholars.In this paper,we use variational methods,such as Moser iteration approach,Nehari manifold and monotonicity trick to obtain the existence and multiplicity of solutions to quasilinear Schrodinger equation.The thesis consists of four sections.In chapter 1,we introduce the background and current situation of quasilinear Schrodinger equation.In Chapter 2,the following quasilinear Schrodinger equation is discussed,where N?3,p?(3,(N+2)/(N-2))and the potentials V satisfies(V)V is 1-periodic,V ? C(RN,R),0<a?inf x?RN V(x),where a is a positive constant.To obtain the existence of solutions to the above equation,by Nehari manifold methods,we firstly consider the existence of the ground state solutions to the perturbed quasilinear Schrrodinger equations and then,letting ??0 and using Moser iteration theory,the non-trivial solution to the original equation is obtained.In Chapter 3,by using a revised Clark theorem and combining with the priori estimate of the solution,we obtain the existence of infinitely many nontrivial solutions to the following quasilinear Schrodinger equation-?u+V(x)u-?[(1+u2)1/2]u/2(1+u2)1/2=K(x)f(u),x?RN,where N?3 and the potentials V and K and the nonlinear term f satisfy(V)0<??V(x)??<?,x?RN;(f1)f is odd near zero,and there exists q E(1,2),such that(f2)(?)=+?,where F(t)=?0t f(s)ds;(K)K(?)0 and K?L2/2-q(RN)?L?(RN).In Chapter 4,the following class of quasilinear Schrodinger equation-?u+V(x)u+k/2[?(u2)]u=?l(u),x?RN,is studied,where N?3,?>0 and K?G R.With the appropriate assumptions on the potential V and local assumptions on the nonlinear term l,we obtain the existence of positive solutions to the above equation using monotonicity trick developed by Jeanjean and minimax methods with careful L?-estimates.Specifically,the potentials V and the nonlinearity l satisfy the following conditions.(V0)V?C1(RN,R)is radially symmetric and there exist a0,a1>0 such that 0<a0?V(x)?a1<? for all x?RN;(Vi)there exists the positive constants C±>0 such that|?V(x)·x|?C±/|x|2,x?RN\{0}where C+<(N-2)2/2 for k>0 and C-<for k<0 respectively;(l1)l?C1(R,R),l(t)=0 for t?0 and there exists q?(2,2*)such that(l2)there exists p?(2,2*)such that(?)L(t)/tp>0,where L(t)=l(s)ds.