Existence and uniqueness theorems for some semi-linear equations on locally finite graphs
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- by Andrea Pinamonti and Giorgio Stefani
- Proc. Amer. Math. Soc. 150 (2022), 4757-4770
- DOI: https://doi.org/10.1090/proc/16046
- Published electronically: August 5, 2022
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Abstract:
We study some semi-linear equations for the $(m,p)$-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all $m\in \mathbb {N}$ and $p\in (1,+\infty )$ via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When $m=1$, we also establish a uniqueness result in the spirit of the Brezis–Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan–Warner-type equations on locally finite weighted graphs.References
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Bibliographic Information
- Andrea Pinamonti
- Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
- MR Author ID: 997336
- Email: andrea.pinamonti@unitn.it
- Giorgio Stefani
- Affiliation: Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste (TS), Italy
- MR Author ID: 1194780
- ORCID: 0000-0002-1592-8288
- Email: giorgio.stefani.math@gmail.com
- Received by editor(s): November 14, 2021
- Published electronically: August 5, 2022
- Additional Notes: The authors are members of INdAM–GNAMPA. The first author was supported by the INdAM–GNAMPA Project 2020 Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali. The second author was partially supported by the ERC Starting Grant 676675 FLIRT – Fluid Flows and Irregular Transport, by the INdAM–GNAMPA Project 2020 Problemi isoperimetrici con anisotropie (n. prot. U-UFMBAZ-2020-000798 15-04-2020), by the INdAM–GNAMPA 2022 Project Analisi geometrica in strutture subriemanniane, codice CUP_E55-F22-00-02-70-001, and has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 945655).
- Communicated by: Nageswari Shanmugalingam
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4757-4770
- MSC (2020): Primary 35R02; Secondary 35J91, 35A15
- DOI: https://doi.org/10.1090/proc/16046
- MathSciNet review: 4489310