On a class of quasilinear elliptic equation with indefinite weights on graphs

TL;DR

This paper investigates a class of quasilinear elliptic equations with indefinite weights on graphs, establishing eigenvalue properties and positive solutions using variational methods, extending the understanding of such equations in discrete settings.

Contribution

It introduces new results on the existence and properties of eigenvalues and positive solutions for quasilinear elliptic equations with indefinite weights on graphs.

Findings

Existence and monotonicity of the principal eigenvalue.

Existence of positive solutions via variational methods.

Extension of elliptic PDE theory to graph-based equations.

Abstract

Suppose that $G=(V, E)$ is a connected locally finite graph with the vertex set $V$ and the edge set $E$. Let $\Omega\subset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ \left \{ \begin{array}{lcr} -\Delta_{p}u= \lambda K(x)|u|^{p-2}u+f(x,u), \ \ x\in\Omega^{\circ}, u=0, \ \ x\in\partial \Omega, \\ \end{array} \right. $$ where $\Omega^{\circ}$ and $\partial \Omega$ denote the interior and the boundary of $\Omega$ respectively, $\Delta_{p}$ is the discrete $p$-Laplacian, $K(x)$ is a given function which may change sign, $\lambda$ is the eigenvalue parameter and $f(x,u)$ has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.