Uniform Bound of Solutions of Chern–Simons–Higgs Equations on Locally Finite Graphs

In this paper, we prove that any bounded solution $$u$$ of the Chern–Simons–Higgs type equation $$-\Delta_p u=u^2(1-u^2)u-\dfrac{1}{2}(1-u^2)^2u\quad\text{in}\ \ V$$ satisfies $$|u|\leq 1$$ , where $$V$$ is a locally finite graph and $$\Delta_p $$ is the $$p$$ -Laplacian on $$V$$ , $$p>1$$ . We will show that the boundedness assumption is necessary by giving a counter-example. Moreover, we also obtain analogue results for the Chern–Simons–Higgs type system $$\begin{cases} -\Delta_p u=u^2(1-u^2-\gamma v^2)u-\dfrac{1}{2}(1-u^2-\gamma v^2)^2u\quad\text{in}\ \ V, \\[10pt] -\Delta_p v=v^2(1-v^2-\gamma u^2)v-\dfrac{1}{2}(1-v^2-\gamma u^2)^2v\quad\text{in}\ \ V, \end{cases}$$ where $$\gamma>0$$ .