Chern-Simons Higgs models for p-Laplacian on finite graphs: a topological degree approach

TL;DR

The paper studies Chern-Simons Higgs models for the $p$-Laplacian on finite graphs using topological degree theory. It first establishes a priori bounds to control solutions and then develops auxiliary variational and monotone-iteration frameworks to handle the nonlinear $p$-Laplacian, addressing both signs of $λ$. By constructing a homotopy and analyzing the associated degree, the authors prove existence of solutions to the graph-based CS-Higgs model under the condition $λ∫ f dμ<0$. This approach extends degree-theoretic methods to nonlinear graph PDEs and provides a robust tool for solvability in discrete geometric settings.

Abstract

We investigate the Chern-Simons Higgs models for p-Laplacian on a connected finite graph, employing topological degree theory as our main tool. Notably, we overcome the difficulties arising from the nonlinearity of p-Laplacian operator and calculate the corresponding topological degree through a more general approach.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Theorem \ref{['t1.1']}
• Lemma 3.1
• proof