The variational method and critical point theory are employed to investigate the existence of solutions for 2 n th-order difference equation Δ n ( p k − n Δ n y k − n ) + ( − 1 ) n + 1 f ( k , y k ) = 0 for k ∈ [ 1 , N ] with boundary value condition y 1 − n = y 2 − n = ⋯ = y 0 = 0 ,   y N+1 = ⋯ = y N+n = 0 by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.