Existence and multiplicity results for boundary value problems connected   with the discrete p(.)-Laplacian on weighted finite graphs
  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. 
Marek Galewski and Renata Wieteska
math.CA