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arXiv · 2207.05445

Quasi-Linear Criticality Theory and Green's Functions on Graphs

Florian Fischer
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Abstract

We study energy functionals associated with quasi-linear Schr\"odinger operators on infinite graphs, and develop characterisations of (sub-)criticality via Green's functions, harmonic functions of minimal growth and capacities. We proof a quasi-linear version of the Agmon-Allegretto-Piepenbrink theorem, which says that the energy functional is non-negative if and only if there is a positive superharmonic function. Furthermore, we show that a Green's function exists if and only if the energy functional is subcritical. Comparison principles and maximum principles are the main tools in the proofs.

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Florian Fischer. 2022-07-12. Quasi-Linear Criticality Theory and Green's Functions on Graphs. https://arxiv.org/abs/2207.05445

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