20.500.12556/RUL-145680 Elliptic problems on weighted locally finite graphs 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. semi-linear equations on graphs variational methods critical point theory true false true Angleški jezik Ni določen Članek v reviji 2023-05-08 08:58:38 2023-05-08 08:58:45 2023-06-28 11:01:54 0000-00-00 00:00:00 2023 0 0 Str. 501-526 no. 1 Vol. 61 Mar. 2023 0000-00-00 PostprintKoncna Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 2024-03-01 517.956 1230-3429 10.12775/TMNA.2022.059 144526083 RAZ_Imbesi_Maurizio_2023.pdf RAZ_Imbesi_Maurizio_2023.pdf 1 E67DA04FAA0F3D6F12B4A8EC92BD2E16 1722180fd072de8d153de5e41aecebdfc61f12aee9c7f2647940bff33b173da1 295882b5-ed6f-11ed-a314-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=168649 https://apcz.umk.pl/TMNA/article/view/42930 1 8d871b93-ed6d-11ed-a314-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=168647 Pedagoška fakulteta Fakulteta za matematiko in fiziko 0 0 0