Opuscula Math. 36, no. 3 (2016), 315-336
http://dx.doi.org/10.7494/OpMath.2016.36.3.315

 
Opuscula Mathematica

Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Majda Chaieb
Abdelwaheb Dhifli
Samia Zermani

Abstract. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{n}\) (\(n\geq 2\)) with a smooth boundary \(\partial \Omega\). We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system \[\begin{aligned} -\Delta u&=a_{1}(x)u^{\alpha}v^{r}\quad\text{in}\;\Omega ,\;\;\,u|_{\partial\Omega}=0,\\ -\Delta v&=a_{2}(x)v^{\beta}u^{s}\quad\text{in}\;\Omega ,\;\;\,v|_{\partial\Omega }=0.\end{aligned}\] Here \(r,s\in \mathbb{R}\), \(\alpha,\beta \lt 1\) such that \(\gamma :=(1-\alpha)(1-\beta)-rs\gt 0\) and the functions \(a_{i}\) (\(i=1,2\)) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory.

Keywords: semilinear elliptic system, asymptotic behavior, Karamata class, sub-super solution.

Mathematics Subject Classification: 31B25, 34B15, 34B18, 34B27.

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  • Majda Chaieb
  • Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
  • Abdelwaheb Dhifli
  • Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
  • Samia Zermani
  • Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2014-06-10.
  • Revised: 2015-10-27.
  • Accepted: 2015-11-15.
  • Published online: 2016-02-21.
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Cite this article as:
Majda Chaieb, Abdelwaheb Dhifli, Samia Zermani, Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain, Opuscula Math. 36, no. 3 (2016), 315-336, http://dx.doi.org/10.7494/OpMath.2016.36.3.315

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