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In this paper we probe into the existences of semilinear discrete Schrodinger equa-tions.In the first chapter, we introduce the background of the discrete Schrodinger equa-tions, some revelent definitions and notes.The contents are as follows:In the second chapter, we study the ground state of the following discrete Schrodinger equations with superlinear nonlinearties: whereΔun=un+1+un-1-2un,Δvn=vn+1+vn-1-2vn,Δis the discrete Laplacian in one spatial dimension. The given sequences en and f(n, s), g(n, s) are assumed to beκ-periodic in n. f(n,s), g(n, s) are assumed to be superlinear at infinity. With the help of Zou[1]’s weak linking theorem, we are able to obtain the existences of the ground state solutions of the above Schrodinger equations without the famous Ambrosetti-Rabinowitz condition. The results proved in our paper improve and extend the results of A. Pankov[2], Y. H. Ding[3], H. He[4], Gongbao Li[5] and so on and the method is not similar to them. In the third chapter, we study the following discrete Schrodinger equations with asymptotically nonlinearties whereΔun=un+1+un-1-2un,Δvn=vn+1+vn-1-2vn,Δis the discrete Laplacian in one spatial dimension, the given sequencesεn are sign-changing and f(n,s), g(n,s) are asymptotically linear at infinity. With the help of Gongbao Li[6]’s weak linking theorem, we are able to obtain the existences of solu-tions of the above Schrodinger equations when 0 does not lie in the spectral gap of the operator -Δ+εn. The results proved in our paper improve and extend the results of A. Pankov[7],Y. H. Ding[8], H. He[9], Gongbao Li[6] and the method is not similar to them.
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