Quasilinear Schr\"odinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions

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.