  In this paper we consider the following quasilinear Schr\"odinger-Poisson
system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll}
- \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi -
\varepsilon^{4}\Delta_4 \phi = u^{2} & \ \mbox{in } \Omega, u=\phi=0 & \
\mbox{on } \partial\Omega \end{array}
  \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$
is assumed to have critical exponencial growth. We first prove existence of
nontrivial solutions $(u_{\varepsilon}, \phi_{\varepsilon})$ and then we show
that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial
solution of the associated Schr\"odinger-Poisson system, that is by making
$\varepsilon=0$ in the system above.
