Generalized Schrödinger-Poisson type systems

Antonio Azzollini; Pietro d’Avenia; Valeria Luisi

doi:10.3934/cpaa.2013.12.867

Article Contents

2013, Volume 12, Issue 2: 867-879. Doi: 10.3934/cpaa.2013.12.867

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Generalized Schrödinger-Poisson type systems

1.
Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell'Ateneo Lucano 10, I-85100 Potenza
2.
Dipartimento di Matematica, Politecnico di Bari, Via Orabona, 4, I-70125 Bari
3.
Dipartimento di Matematica, Universita degli Studi di Bari, Via E. Orabona 4, 70125 Bari

Received: August 31, 2011

Revised: May 31, 2012

Published: February 2013

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Abstract

In this paper we study the boundary value problem \begin{eqnarray*} -\Delta u+ \varepsilon q\Phi f(u)=\eta|u|^{p-1}u \quad in \quad \Omega, \\ - \Delta \Phi=2 qF(u) \quad in \quad \Omega, \\ u=\Phi=0 \quad on \quad \partial \Omega, \end{eqnarray*} where $\Omega \subset R^3$ is a smooth bounded domain, $1 < p < 5$, $\varepsilon,\eta= \pm 1$, $q>0$, $f: R\to R$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.

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Mathematics Subject Classification: Primary: 35J55, 35J60; Secondary: 35J20.

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