The Existence Of Homoclinic Solutions For Discrete Nonlinear Schr(?)dinger Equations

In this paper, we mainly concern with the existence of homoclinic solutions for two classes of discrete nonlinear Schrodinger equations. First, we establish some suit-able variational functionals, such that homoclinic solutions corresponded to critical points of the functionals. Then we obtain sufficient conditions on the existence of homoclinic solutions of the DNLS equation by using the mountain pass theorem (MPT) and the fountain theorem. This thesis is organized as follows:In chapter1, we concentrate on the introduction of background and applications of Schrodinger equations, existed results of this field and main work in this thesis.In chapter2, we give some basic knowledge which is very useful in this thesis.In chapter3, we mainly consider the existence and multiplicity results of homo-clinic solutions for the following DNLS equation Lun+vnun-ωun=σf(n,un), n∈Zm, with unbounded potentials. V={vn} satisfies:(?) vn=∞. The nonlinearity f is supposed to satisfy the following assumptions.(f1)f∈C(Zm×R,R), and there exist a>0, p∈(2,∞) such that|f(n,u)|≤a(1+|u|p-1), for all n∈Zm,u∈R.(f2)(?) f(n,u)/u=0uniformly for n∈Zm.(f3)(?) F(n, u)/u2=+∞uniformly for n∈Zm, where F(n, u) is the primitive function of f(n, u).(f4) f(n,u)/u is increasing in u>0and decreasing in u<0for all n∈Zm.The proof is based on the mountain pass theorem and the fountain theorem.In chapter4, we mainly research the existence of the homoclinic solution for the folloing DNLS equation Lun+vnun-ωun=γn|un|P-2un,n∈Znwith anharmonic parameters.where γn denotes an anharmonic parameter.The proof is based on the mountain pass theorem.