Multiple solutions of a nonlinear biharmonic equation on graphs

TL;DR

This paper investigates a nonlinear biharmonic equation on graphs, demonstrating the existence of multiple solutions using variational methods within a graph-theoretic Dirichlet problem setting.

Contribution

It introduces a novel approach to solving nonlinear biharmonic equations on graphs and proves the existence of multiple solutions under specific conditions.

Findings

Proved existence of two distinct solutions for the biharmonic equation on graphs.

Applied variational methods to a graph-based Dirichlet problem.

Extended classical PDE techniques to discrete graph settings.

Abstract

In this paper, we consider a biharmonic equation with respect to the Dirichlet problem on a domain of a locally finite graph. Using the variation method, we prove that the equation has two distinct solutions under certain conditions.