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Existence of solutions for a semilinear ellipticsystem

Published online by Cambridge University Press:  15 February 2013

Mohamed Benrhouma
Mohamed Benrhouma*
Affiliation:
Mathematics Department, Sciences Faculty of Monastir, 5019 Monastir, Tunisia. [email protected]
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Abstract

This paper deals with the existence of solutions to the following system:

$$\left\{\begin{array}{l} -\Deltau+u=\frac{\alpha}{\alpha+\beta}a(x)|v|^{\beta} |u|^{\alpha-2}u\quad\mbox{ in}\mathbb{R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta}{\alpha+\beta}a(x)|u|^{\alpha}|v|^{\beta-2}v\quad\mbox{ in }\mathbb{R}^N. \end{array}\right.$$−Δu+u=αα+βa(x)|v|β|u|α−2u inRN−Δv+v=βα+βa(x)|u|α|v|β−2v inRN.

With the help of the Nehari manifold and the linking theorem, we prove the existence ofat least two nontrivial solutions. One of them is positive. Our main tools are theconcentration-compactness principle and the Ekeland’s variational principle.

Keywords

Semilinear elliptic systemsNehari manifoldconcentration-compactness principlevariational methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 574 - 586
Copyright
© EDP Sciences, SMAI, 2013

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