Nontrivial Solutions For A3-dimensional Discrete Non-cooperative Systems With Resonance

The purpose of this paper is to study the existence of the nontrivial solutions for the3-dimensional discrete non-cooperative resonant systems by using the variational method of the nonlinear functional analysis, the critical point theory, especially the critical groups and the Morse theory. The system is as follows: where N(≥3) is a given integer, discrete interval [1,N]={1,2,…,N},Δ denotes the forward difference operator which defined by Δu(k)=u(k+1)-u(k), Δ2u(k)=Δ(Δu(k)), and nonlinear term F∈C2(R3,R1) such thatF(0)=0,▽(0)=0,▽F denotes the gradient of F.This paper consists of the following four chapters:In chapter one, we introduce the forefront and the significance of the study about system (1.2.1) by variational method and give the main results, i.e. Theorems1.2.1-1.2.3. Through Theorems1.2.1-1.2.2, we describe the characters of eigenvalues for the linear eigenvalue systems corresponding to system (1.2.1). Then we obtain four sufficient conditions for the existence of the nontrival solutions for system (1.2.1) by Theorem1.2.3In chapter two, we give some basic knowledge and relevant results of the critical point theory, which is needed in the textIn chapter three, we give the proofs of Theorems1.2.1-1.2.2by the matrix theory. Still, we get the matrix form and the energy functional of system (1.2.1) and describe some characters of the energy functional by the eigenvalues of the linear eigenvalue systems. These characters will be used in the proofs of Theorem1.2.3.In chapter four, based on the computations of the critical groups, the Morse theory and the relevant conclusions drawn by chapter two and chapter three, we get the proofs of Theorem1.2.3.