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Existence of solutions for a semilinear elliptic system

Published online by Cambridge University Press:  15 February 2013

Mohamed Benrhouma
Mohamed Benrhouma*
Affiliation:
Mathematics Department, Sciences Faculty of Monastir, 5019 Monastir, Tunisia. brhouma06@yahoo.fr
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Abstract

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

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

With the help of the Nehari manifold and the linking theorem, we prove the existence of at least two nontrivial solutions. One of them is positive. Our main tools are the concentration-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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