Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems Δ12u(i−1,j)+Δ22u(i,j−1)=−f((i,j),u(i,j)),(i,j)∈Ω,u(i,0)=u(i,T2+1)=0i∈Z(1,T1),u(0,j)=u(T1+1,j)=0j∈Z(1,T2), which have a symmetric structure. When the nonlinearity f(·,u) is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking.