Existence of Nontrivial Solutions for the Nonlinear Equation on Locally Finite Graphs

Abstract

Suppose that $G=(V,E)$ be a locally finite and connected graph with symmetric weight and uniformly positive measure, where $V$ denotes the vertex set and $E$ denotes the edge set. We are concered with the following problem $$\{\begin{array}{ll}-\Delta u+hu=f(x,u),& \text{ in }\Omega ,\\ u=0,& \text{ on }\partial \Omega ,\end{array}$$ on the graph, where $h:\Omega \to \mathbb{R}$, $f:\Omega \times \mathbb{R}\to \mathbb{R}$ and $u:\Omega \to \mathbb{R}$. When $f$ and $h$ satisfies certain assumption conditions, we can ascertain the existence of one or two nontrivial solutions on the graph.

A total of 21 pages