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Elliptic problems on weighted locally finite graphs
ID Imbesi, Maurizio (Avtor), ID Molica Bisci, Giovanni (Avtor), ID Repovš, Dušan (Avtor)

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Izvleček
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.

Jezik:Angleški jezik
Ključne besede:semi-linear equations on graphs, variational methods, critical point theory
Vrsta gradiva:Članek v reviji
Tipologija:1.01 - Izvirni znanstveni članek
Organizacija:PEF - Pedagoška fakulteta
FMF - Fakulteta za matematiko in fiziko
Status publikacije:Objavljeno
Različica publikacije:Recenzirani rokopis
Leto izida:2023
Št. strani:Str. 501-526
Številčenje:Vol. 61, no. 1
PID:20.500.12556/RUL-145680 Povezava se odpre v novem oknu
UDK:517.956
ISSN pri članku:1230-3429
DOI:10.12775/TMNA.2022.059 Povezava se odpre v novem oknu
COBISS.SI-ID:144526083 Povezava se odpre v novem oknu
Datum objave v RUL:08.05.2023
Število ogledov:277
Število prenosov:4
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Gradivo je del revije

Naslov:Topological Methods in Nonlinear Analysis
Skrajšan naslov:Topol. Methods Nonlinear Anal.
Založnik:Juliusz Schauder Center for Nonlinear Studies
ISSN:1230-3429
COBISS.SI-ID:14203653 Povezava se odpre v novem oknu

Licence

Licenca:CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
Povezava:http://creativecommons.org/licenses/by/4.0/deed.sl
Opis:To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Projekti

Financer:Drugi - Drug financer ali več financerjev
Program financ.:Italy, MIUR
Številka projekta:2015KB9WPT 009
Naslov:Variational methods, with applications to problems in mathematical physics and geometry

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:P1-0292
Naslov:Topologija in njena uporaba

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:N1-0114
Naslov:Algebrajski odtisi geometrijskih značilnosti v homologiji

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:N1-0083
Naslov:Forsing, fuzija in kombinatorika odprtih pokritij

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