Existence of solutions for nonlinear $p$-Laplacian difference equations

The aim of this paper is the study of existence of solutions for nonlinear $2n^{\mathrm{th}}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.