Quasi-linear Schrödinger–Poisson system under an exponential critical nonlinearity: existence and asymptotic behaviour of solutions

Abstract

In this paper we consider the following quasilinear Schrödinger–Poisson system in a bounded domain in ${\mathbb {R}}^{2}$:

$$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &{}\ \text{ in } \Omega , \\ -\Delta \phi - \varepsilon ^{4}\Delta _4 \phi = u^{2} &{} \ \text{ in } \Omega ,\\ u=\phi =0 &{} \ \text{ on } \partial \Omega \end{array} \right. \end{aligned}$$

depending on the parameter $\varepsilon >0$. The nonlinearity f is assumed to have critical exponential growth. We first prove existence of nontrivial solutions $(u_{\varepsilon }, \phi _{\varepsilon })$ and then we show that as $\varepsilon \rightarrow 0^{+}$, these solutions converge to a nontrivial solution of the associated Schrödinger–Poisson system, that is, by making $\varepsilon =0$ in the system above.