Abstract:
In this paper, we prove that any bounded solution
$u$
of the
Chern–Simons–Higgs type equation
\begin{equation*}
-\Delta_p u=u^2(1-u^2)u-\dfrac{1}{2}(1-u^2)^2u\quad\text{in}\ \ V
\end{equation*}
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{equation*}
\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}
\end{equation*}
where
$\gamma>0$.
Keywords:Chern–Simons–Higgs equations, Boundedness of solutions,
$p$-Laplace operator,
locally finite graph.
