Fractional Chern–Simons–Higgs models on finite graphs via a topological degree approach

We investigate the fractional Chern–Simons–Higgs models of the form ${(-\Delta )}^{s}u=\lambda e^u{(e^u-1)}^{2p+1}+f$on a connected finite graph, where ${(-\Delta )}^s$ is the fractional Laplace operator, $s\in (0,1)$, $\lambda$ is a real number, $p$ is a non-negative integer, and $f$ is a function on the graph. We focus on the fractional Laplace operator defined with heat kernels and employ the topological degree theory as our main tool. First, we prove that all solutions to fractional Chern–Simons–Higgs models are uniformly bounded. Second, we calculate the topological degree by discussing the existence of the solution to a homotopy equation case by case. As consequences, we obtain the existence results for the fractional Chern–Simons–Higgs models on a connected finite graph.