Existence and convergence of solutions for nonlinear elliptic systems on graphs

TL;DR

This paper studies nonlinear elliptic systems on graphs, proving existence, convergence, and concentration behavior of solutions using variational methods, and supports findings with numerical experiments.

Contribution

It establishes the existence and convergence of solutions for nonlinear elliptic systems on graphs, including their concentration behavior as parameters vary.

Findings

Existence of nontrivial ground state solutions via mountain pass theorem.

Solutions concentrate and converge to Dirichlet problem solutions as parameter increases.

Numerical experiments validate theoretical results.

Abstract

We consider a kind of nonlinear systems on a locally finite graphs $G=(V,E)$. We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solution which depends on the parameter $\lambda$ with some suitable assumptions on the potentials. Moreover, we pay attention to the concentration behavior of these solutions and prove that, as $\lambda \to \infty$, these solutions converge to a ground state solution of a corresponding Dirichlet problem. Finally, we also provide some numerical experiments to illustrate our results.