Existence of solutions for nonlinear p-Laplacian diference equations

TL;DR

This paper investigates the existence of solutions for nonlinear discrete p-Laplacian difference equations, proving the existence of homoclinic solutions and multiple solutions using variational methods.

Contribution

It introduces new existence results for solutions of nonlinear p-Laplacian difference equations, employing mountain-pass and critical point theorems.

Findings

Existence of a nontrivial homoclinic solution proven

Multiple solutions for boundary value problems established

Application of variational methods to discrete p-Laplacian equations

Abstract

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.