Existence and multiplicity results for boundary value problems connected with the discrete p(.)-Laplacian on weighted finite graphs

TL;DR

This paper investigates the existence and multiplicity of positive solutions for boundary value problems involving the discrete p(.)-Laplacian on weighted finite graphs using variational methods and auxiliary inequalities.

Contribution

It introduces new variational techniques and auxiliary inequalities to establish existence and multiplicity results for discrete p(.)-Laplacian boundary value problems on finite graphs.

Findings

Existence of positive solutions under certain conditions

Multiple solutions demonstrated via variational methods

New inequalities for the discrete p(.)-Laplacian derived

Abstract

We use the direct variational method, the Ekeland variational principle, the mountain pass geometry and Karush-Kuhn-Tucker theorem in order to investigate existence and multiplicity results for boundary value problems connected with the discrete p(.)-Laplacian on weighted finite graphs. Several auxiliary inequalities for the discrete p(.)-Laplacian on finite graphs are also derived. Positive solutions are considered.