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

TL;DR

This paper develops a ground state representation for non-local quasi-linear Schr"odinger operators, establishing a criticality theory and Hardy inequality characterizations on graphs and Euclidean spaces, with applications to Liouville principles.

Contribution

It introduces a novel ground state representation and criticality theory for non-local quasi-linear Schr"odinger operators on graphs and Euclidean spaces.

Findings

Established a criticality theory for these operators.

Provided characterizations for Hardy inequalities.

Proved a Liouville comparison principle on graphs.

Abstract

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.