A Non-Local Quasi-Linear Ground State Representation and Criticality Theory

We study energy functionals associated with non-local quasi-linear Schr\"odinger operators, and develop a ground state representation. Our main focus is on infinite graphs but we also consider non-local quasi-linear Schr\"odinger operators in the Euclidean space. Using the representation, we develop a criticality theory for quasi-linear Schr\"odinger operators on general weighted graphs, and show characterisations for a Hardy inequality to hold true. As an application, we show a Liouville comparison principle on graphs.