Topological degree for Chern-Simons Higgs models on finite graphs

Abstract

Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\hat{I}”u=\hat{I}»e^u(e^u-1)+f,(0.1)$$ where $\hat{I}”$ is the graph Laplacian, $\hat{I}»$ is a real number and $f$ is a function on $V$. When $\hat{I}»>0$ and $f=4\ddot{I}€{\sum }_{i=1}^N\hat{I}{´}_{p_i}$, $N\in \mathbb{N}$, $p_1,\cdots ,p_N\in V$, the equation (0.1) was investigated by Huang, Lin, Yau (Commun. Math. Phys. 377 (2020) 613-621) and Hou, Sun (Calc. Var. 61 (2022) 139) via the upper and lower solutions principle. We now consider an arbitrary real number $\hat{I}»$ and a general function $f$, whose integral mean is denoted by $\overline{f}$, and prove that when $\hat{I}»\overline{f}<0$, the equation $(0.1)$ has a solution; when $\hat{I}»\overline{f}>0$, there exist two critical numbers $\hat{I}{›}^{\ast }>0$ and $\hat{I}{›}_{\ast }<0$ such that if $\hat{I}»\in (\hat{I}{›}^{\ast },+\infty )\cup (-\infty ,\hat{I}{›}_{\ast })$, then $(0.1)$ has at least two solutions, including one local minimum solution; if $\hat{I}»\in (0,\hat{I}{›}^{\ast })\cup (\hat{I}{›}_{\ast },0)$, then $(0.1)$ has no solution; while if $\hat{I}»=\hat{I}{›}^{\ast }$ or $\hat{I}{›}_{\ast }$, then $(0.1)$ has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern-Simons Higgs system, and a partial result for the multiple solutions of the system is obtained.

20 pages