Existence of solutions for $p$-Laplacian discrete equations

TL;DR

This paper investigates the existence of solutions for discrete p-Laplacian equations using variational methods, providing new existence results for second-order discrete problems with parameter dependence.

Contribution

It establishes an existence theorem for solutions to discrete p-Laplacian equations, including a specific case involving a second-order problem with a parameter, using variational techniques.

Findings

Existence of at least one non-zero solution proven.

Derived an existence theorem for a second-order discrete problem.

Applicable variational methods in finite-dimensional setting.

Abstract

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a positive real parameter $\alpha$, whose prototype is given by $$ \left\{ \begin{array}{l} -\Delta^2u({k-1})=\alpha f(k,u(k)),\quad \forall\; k \in {\mathbb{Z}}[1,T] \\ {u(0)=u({T+1})=0.}\\ \end{array} \right. $$ Our approach is based on variational methods in finite-dimensional setting.