The Kazdan–Warner equation on canonically compactifiable graphs

Abstract

We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

Sobolev-type inequalities and eigenvalue growth on graphs with finite measure

“…There it was shown that that Laplacians on these graphs share indeed many properties with Laplacians on pre-compact manifolds with nice boundary. In [KS18], non-linear equations were studied on these graphs using the condition including the measure as we have introduced it above. For Dirichlet forms Q 1 , Q 2 we write Q 1 ≥ Q 2 if D(Q 1 ) ⊆ D(Q 2 ) and Q 1 (u) ≥ Q 2 (u) for every u ∈ D(Q 1 ).…”

Existence of Solutions to Mean Field Equations on Graphs

“…In [5], Grigorigan, Lin and Yang have obtained a few sufficient conditions when equation (1.2) has a solution on a finite graph. There are several further results regarding the solutions of (1.2) on graphs in [6], [7], [8].…”

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

“…For example, when domains are finite graphs, they proved existence of solutions for the Kazdan-Warner equation [10] and the Yamabe type equation [11]. Later, results in [10,11] were generalized by Ge and Jiang [8,9] for infinite graphs and Keller and Schwarz [18] also studied the Kazdan-Warner equation on canonically compact graphs.…”