  In this paper we consider the following quasilinear Schr\"odinger-Poisson
system $$ \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda
f(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -\Delta \phi
-\varepsilon^{4} \Delta_4 \phi = u^{2} & \ \mbox{in } \mathbb{R}^{3},
\end{array}
  \right. $$ depending on the two parameters $\lambda,\varepsilon>0$.
  We first prove that, for $\lambda$ larger then a certain $\lambda^{*}>0$,
there exists a solution for every $\varepsilon>0$. Later, we study the
asymptotic behaviour of these solutions whenever $\varepsilon$ tends to zero,
and we prove that they converge to the solution of the Schr\"odinger-Poisson
system associated.
