Multiple homoclinic solutions for indefinite second-order discrete Hamilton system with small perturbation

In this article, we sutdy the multiplicity of homoclinic solutions to the perturbed second-order discrete Hamiltonian system $$ \Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\nabla W(n,u(n))+\theta\nabla F(n,u(n))=0, $$ where L(n) and W(n,x) are neither autonomous nor periodic in n. Under the assumption that W(n,x) is only locally superquardic as $|x|\to \infty$ and even in x and F(n,x) is a perturbation term, we establish some existence criteria to guarantee that the above system has multiple homoclinic solutions by minimax method in critical point theory