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

Abstract

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 $$u\equiv 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.

Topics & Concepts