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Elliptic problems on weighted locally finite graphs
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Imbesi, Maurizio
(
Author
),
ID
Molica Bisci, Giovanni
(
Author
),
ID
Repovš, Dušan
(
Author
)
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https://apcz.umk.pl/TMNA/article/view/42930
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Abstract
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.
Language:
English
Keywords:
semi-linear equations on graphs
,
variational methods
,
critical point theory
Work type:
Article
Typology:
1.01 - Original Scientific Article
Organization:
PEF - Faculty of Education
FMF - Faculty of Mathematics and Physics
Publication status:
Published
Publication version:
Author Accepted Manuscript
Year:
2023
Number of pages:
Str. 501-526
Numbering:
Vol. 61, no. 1
PID:
20.500.12556/RUL-145680
UDC:
517.956
ISSN on article:
1230-3429
DOI:
10.12775/TMNA.2022.059
COBISS.SI-ID:
144526083
Publication date in RUL:
08.05.2023
Views:
505
Downloads:
43
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Record is a part of a journal
Title:
Topological Methods in Nonlinear Analysis
Shortened title:
Topol. Methods Nonlinear Anal.
Publisher:
Juliusz Schauder Center for Nonlinear Studies
ISSN:
1230-3429
COBISS.SI-ID:
14203653
Licences
License:
CC BY 4.0, Creative Commons Attribution 4.0 International
Link:
http://creativecommons.org/licenses/by/4.0/
Description:
This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
Projects
Funder:
Other - Other funder or multiple funders
Funding programme:
Italy, MIUR
Project number:
2015KB9WPT 009
Name:
Variational methods, with applications to problems in mathematical physics and geometry
Funder:
ARRS - Slovenian Research Agency
Project number:
P1-0292
Name:
Topologija in njena uporaba
Funder:
ARRS - Slovenian Research Agency
Project number:
N1-0114
Name:
Algebrajski odtisi geometrijskih značilnosti v homologiji
Funder:
ARRS - Slovenian Research Agency
Project number:
N1-0083
Name:
Forsing, fuzija in kombinatorika odprtih pokritij
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