Elliptic problems on weighted locally finite graphs

Imbesi, Maurizio (avtor), Molica Bisci, Giovanni (avtor), Repovš, Dušan (avtor)

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
• UDK: 517.956
• ISSN pri članku: 1230-3429
• DOI: 10.12775/TMNA.2022.059
• COBISS.SI-ID: 144526083
• Datum objave v RUL: 08.05.2023

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

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.