Abstract
In the theory of potentials on Riemannian manifolds, Sario et al. [14] introduced the notion of $(p,q)$-biharmonic functions, i.e., $\Delta_q\Delta_pu = 0$, where $- \Delta_q = q - \Delta$ is a linear Schrödinger operator. They showed $(p,q)$-biharmonic classification of Riemannian manifolds and related facts. We show some analogous results by taking infinite networks for Riemannian manifolds. We obtain some new facts relating to the $(p,q)$-version of the growth of $\Delta_qu$ studied in [12]. Our $(p, q)$-biharmonic projection is analogous to that in [14].
Acknowledgment
The authors thank for the three anonymous referees for many valuable comments. The first referee shows us two recent works related to Schrödinger operator on graphs. The second referee points out several insufficient explanation. We would like to thank the third referee for his detailed comments which improved significantly the presentation of this paper.
Citation
Hisayasu KURATA. Maretsugu YAMASAKI. "$(p, q)$-biharmonic functions on networks." Hokkaido Math. J. 53 (1) 111 - 138, February 2024. https://doi.org/10.14492/hokmj/2022-629
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