Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order self-adjoint discrete Hamiltonian system Δ [ p ( n ) Δ u ( n − 1 ) ] − L ( n ) u ( n ) + ∇ W ( n , u ( n ) ) = 0 has infinitely many homoclinic orbits, where n ∈ Z , u ∈ R N , p , L : Z → R N × N and W : Z × R N → R are no periodic in n. Our conditions on the potential W ( n , x ) are rather relaxed.