The variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equation Δn(pk−nΔnyk−n)+(−1)n+1f(k,yk)=0 for k∈[1,N] with boundary value condition y1−n=y2−n=⋯=y0=0, yN+1=⋯=yN+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.
