Existence and convergence of solutions to $p$-Laplace equations on locally finite graphs

TL;DR

This paper studies the existence and behavior of solutions to nonlinear p-Laplace equations on locally finite graphs, extending previous results by allowing broader conditions on p, the graph, and the nonlinear term.

Contribution

It establishes existence and asymptotic properties of solutions for all p in (1, ∞), with relaxed conditions on the graph and nonlinearities, advancing the mathematical understanding of p-Laplace equations on graphs.

Findings

Existence of positive solutions using mountain-pass theorem.

Existence of positive ground state solutions via Nehari manifold.

Analysis of asymptotic behavior of solutions.

Abstract

We are mainly concerned with the nonlinear $p$-Laplace equation \begin{equation*} -\Delta_pu+\rho|u|^{p-2}u=\psi(x,u) \end{equation*} on a locally finite graph $G=(V,E)$, where $p$ belongs to $(1, +\infty)$. We obtain existence of positive solutions and positive ground state solutions by using the mountain-pass theorem and the Nehari manifold respectively. Moreover, we also analyze the asymptotic behavior for a sequence of positive ground state solutions. Compared with all the existing relevant works, our results have made essential improvements in at least three aspects: $(i)$ $p$ can take any value in $(1, +\infty)$; $(ii)$ the conditions on the graph $G$ and the potential $\rho$ are relaxed; $(iii)$ for the existence of positive solutions, the growth condition in previous works on the nonlinear term $\psi(x,s)$ as $s\rightarrow +\infty$ is removed.