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Elliptic problems on weighted locally finite graphsImbesi, Maurizio ; Molica Bisci, Giovanni, 1975- ; Repovš, Dušan, 1954-Let ▫$\mathcal{G}:= (V,E)$▫ be a weighted locally finite graph whose finite measure ▫$\mu$▫ has a positive lower bound. Motivated by a wide interest in the current literature, in this paper we study ... the existence of classical solutions for a class of elliptic equations involving the ▫$\mu$▫-Laplacian operator on the graph ▫$\mathcal{G}$▫, whose analytic expression is given by ▫$$ \Delta_{\mu} u(x) := \frac{1}{\mu (x)} \sum_{y\sim x} w(x,y) (u(y)-u(x))\quad (\text{for all } x\in V),$$▫ where ▫$w \colon V\times V \rightarrow [0,+\infty)$▫ is a weight symmetric function and the sum on the right-hand side of the above expression is taken on the neighbours vertices ▫$x,y\in V$▫, that is ▫$x\sim y$▫ whenever ▫$w(x,y) > 0$▫. More precisely, by exploiting direct variational methods, we study problems whose simple prototype has the following form ▫$$ \begin{cases} -\Delta_{\mu} u(x)=\lambda f(x,u(x))&\text{for } x \in \mathop D\limits^ \circ,\\ u|_{\partial D}=0, \end{cases}$$▫ where ▫$D$▫ is a bounded domain of ▫$V$▫ such that ▫$\mathop D\limits^ \circ\neq \emptyset$▫ and ▫$\partial D\neq \emptyset$▫, the nonlinear term ▫$f \colon D \times \RR \rightarrow \RR$▫ satisfy suitable structure conditions and ▫$\lambda$▫ is a positive real parameter. By applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two solutions for the treated problems. We emphasize the crucial role played by the famous Ambrosetti-Rabinowitz growth condition along the proof of the main theorem and its consequences. Our results improve the general results obtained by A. Grigor'yan, Y. Lin, and Y. Yang.Source: Topological Methods in Nonlinear Analysis. - ISSN 1230-3429 (Vol. 61, no. 1, Mar. 2023, str. 501-526)Type of material - article, component part ; adult, seriousPublish date - 2023Language - englishCOBISS.SI-ID - 144526083
Author | Imbesi, Maurizio ; Molica Bisci, Giovanni, 1975- ; Repovš, Dušan, 1954- |
Title | Elliptic problems on weighted locally finite graphs |
Publication date | 2023-03-04 |
COBISS.SI-ID | 144526083 |
Publication version in repository | Postprint, author's finally reviewed version, accepted for publication |
Publication licence | Creative Commons Attribution 4.0 International |
Embargo | Postpone publication for public until 2024-03-01 |
Project(s) from which the publication was funded
Title | Acronym | Project ID | Funder |
---|---|---|---|
Topologija in njena uporaba | P1-0292-2022 |
Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije |
|
Algebrajski odtisi geometrijskih značilnosti v homologiji | N1-0114-2019 |
Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije |
|
Forsing, fuzija in kombinatorika odprtih pokritij | N1-0083-2018 |
Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije |
Files that belong to the publication
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https://apcz.umk.pl/TMNA/article/view/42930 |
https://d.cobiss.net/repository/si/files/144526083/119906/Clanek-152.pdf |
https://repozitorij.uni-lj.si/IzpisGradiva.php?id=145680 |
source: Topological Methods in Nonlinear Analysis. - ISSN 1230-3429 (Vol. 61, no. 1, Mar. 2023, str. 501-526)
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Database name | Field | Year |
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Links to authors' personal bibliographies | Links to information on researchers in the SICRIS system |
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Imbesi, Maurizio | |
Molica Bisci, Giovanni, 1975- | 36991 |
Repovš, Dušan, 1954- | 07083 |
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