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

  • Giovany M. Figueiredo
  • Gaetano Siciliano
Article
  • 8 Downloads

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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de Brasília-UNBBrasíliaBrazil
  2. 2.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil

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