Elliptic problems on weighted locally finite graphs - Imbesi, Maurizio; Bisci, Giovanni Molica; Repovš, Dušan D. (Topological Methods in Nonlinear Analysis, 2023)
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 classof elliptic equations involving the $\mu$-Laplacian operator on the graph$\mathcal{G}$, whose analytic expression is given by\begin{equation*}\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),\end{equation*}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{equation*}\label{Np000}\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}\end{equation*}where $D$ is a bounded domain of $V$ such that $\mathop D\limits^ \circ eq \emptyset$ and $\partial D eq \emptyset$, 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 aclassical Pucci-Serrin theorem in addition to a local minimum result for differentiablefunctionals due to Ricceri, we are able to prove the existence of at least two solutionsfor the treated problems. We emphasize the crucial role played by the famous Ambrosetti-Rabinowitz growth condition along the proof of the main theoremand its consequences. Our results improve the general results obtained byA.\ Grigor'yan, Y.\ Lin, and Y.\ Yang in \cite{GLY2}.