Existence Of Solutions For Several Classes Of Differential Equations With Nonlocal Terms

In this thesis,we study several classes of differential equations and differential systems with nonlocal terms and establish the existence and multiplicity of solutions.The thesis is organized as follows:In chapter 1,a brief introduction of the background,current situation and the structure of the dissertation is presented.In chapter 2,we discuss the following Kirchhoff equation-(a+b?|?u|2)?u+V(x)u= f(x,u),x?RN,where N ?3 and a,b>0 are constants.Without any growth condition for the nonlinear term f(x,u)on u at infinity,we obtain a sequence of solutions converging to zero in some Sobolev spaces.These results generalize and improve some existing results in the literature.In chapter 3,we study the following quasilinear Schrodinger equation-?u+V(x)u+k/??(A(|u|?)|u|?-2u=f(x,u),x?RN.where N ?3,a>1 is a constant,k>0 is a parameter,V ? C(RN,R)is periodic in xi,1?i?N and the nonlinear term f(x,u)satisfies some conditions.Using variational methods,we prove the existence of nontrivial solutions.Here we allow that f(x,u)can contain superlinear(even supercritical),asymptotically linear or sublinear nonlinearities,respectively.Our results enlarge or improve some existing results in the literature.In chapter 4,a kind of quasilinear Schrodinger equations and a coupled quasilinear Schrodinger systems are discussed by the dual method.We first establish the existence of a nontrivial solution for the quasilinear Schrodinger equation-?u+V(x)u-?(u2)u=K(x)|u|22*-2u+g(x,u)+h(x),x?RN where N ? 3,V,K,h:RN ? R,g ?C(RN x R,R)and the potential is allowed to be sign-changing.The quasilinear equations are reduced to semilinear ones by employing a change of variables.We prove that the equation possesses a nontrivial solution.We are also interested in the following quasilinear Schrodinger systemwhere N ?3,?>0 is a parameter,Vi(x)is a nonnegative potential,Ki(x)is a bounded positive function,i = 1,2.h1(x,u,v)u and/h2(x,u,v)v are superlinear but subcritical functions.Under some proper conditions,minimax methods are employed to establish the existence of the stand-ing wave solutions for this system provided that s is small enough;for any m ? N,it has m pairs of solutions if ? is small enough.And these solutions(u?,v?)?(0,0)in some Sobolev space as??0.Moreover,we establish the existence of positive solutions when ? = 1.In chapter 5,we discuss the following Schrodinger-Poisson system with singularity and critical termswhere ?(?)RN is a bounded domain with smooth boundary(?)?,N ?3,q,?,?>0 are parame-ters,r ?(0,1)is a constant,f:?×R?R is a continuous function and F(x,u)=?0uf(x,t)dt,(?)(x,u)? ? × R.The above system possesses at least two positive solutions under some suitable conditions.In chapter 6,we summarize our conclusions and innovation points and state the prospects of this thesis.