Archiv der Mathematik

, Volume 112, Issue 3, pp 313–327

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

Article

## 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.

## Keywords

Variational methods Nonlocal problems Schrödinger–Poisson equation Exponential critical growth

## Mathematics Subject Classification

35Q60 35J10 35J50 35J92 35J61

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## Authors and Affiliations

• Giovany M. Figueiredo
• 1
• Gaetano Siciliano
• 2
1. 1.Departamento de MatemáticaUniversidade de Brasília-UNBBrasíliaBrazil
2. 2.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil