  The aim of this paper is the study of existence of solutions for non- linear
p-Laplacian difference equations. 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 theorem of D.
Averna and G. Bonanno, which ensures the existence of three critical points for
a suitable functional.
