A Liouville theorem for elliptic equations with a potential on infinite graphs

2023/04/03 by Stefano Biagi, Biagi, Stefano, Giulia Meglioli +3 · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2304.00786

openalex publication_date 2023/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u≡ 0. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

Cited by

• Decay of Mass of the Solution to the Cauchy Problem of the p-Laplacian with Absorption on Infinite Graphs
• Large time behavior of the solution to the Cauchy problem for the discrete p-Laplacian with density on infinite graphs
• On positive solutions of Lane-Emden equations on the integer lattice graphs
• Blow up results and lifespan estimates for nonlinear damped wave equations on weighted graphs

Related

• Phragmèn-Lindelöf type theorems for elliptic equations on infinite graphs
• Nonexistence results for semilinear elliptic equations on weighted graphs
• Uniqueness in weighted l^p spaces for the Schrödinger equation on infinite graphs
• Uniqueness for the Schrödinger Equation on Graphs with Potential Vanishing at Infinity
• A note on Liouville type equations on graphs