Elliptic problems on weighted locally finite graphs

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

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.

Keywords

semi-linear equations on graphs;variational methods;critical point theory;

Data

Language:	English
Year of publishing:	2023
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956
COBISS:	144526083
ISSN:	1230-3429
Views:	81
Downloads:	1
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Embargo end date (OpenAIRE):	2024-03-01
Pages:	str. 501-526
Volume:	ǂVol. ǂ61
Issue:	ǂno. ǂ1
Chronology:	Mar. 2023
DOI:	10.12775/TMNA.2022.059
ID:	18841254