Abstract
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 \(\gamma>0\) .
