拟线性临界理论与图的格林函数
Quasi-Linear Criticality Theory and Green's Functions on Graphs
摘要：
我们研究能量泛函与准线性薛定谔算子在无限图，并通过格林函数，调和函数的最小增长和能力的（次）临界发展的特征。我们证明了Agmon-Allegretto-Piepenbrink定理的准线性版本，该定理表明能量泛函是非负的当且仅当存在正的超调和函数。此外，我们证明了格林函数存在的充要条件是能量泛函是亚临界的。比较原理和极大值原理是证明的主要工具。
英文摘要：
We study energy functionals associated with quasi-linear Schrödinger 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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